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12th Standard Maths English Medium - Important 5 Mark Question Paper and Answer Key 2022 - 2023 - by Study Materials - Dec 31, 2022 - View & Read

1)
Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).
2)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
3)
Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.
4)
If z₁= 2 + 5i, z₂= -3 - 4i, and z₃= 1 + i, find the additive and multiplicate inverse of z₁, z₂ and z₃
5)
If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.

12th Standard Maths English Medium - Important 3 Mark Question Paper and Answer Key 2022 - 2023 - by Study Materials - Dec 31, 2022 - View & Read

1)
Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).
2)
Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)
3)
Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)
4)
If \(2cos\ \alpha=x+\frac { 1 }{ x } \) and \(2cos\ \beta =y+\frac { 1 }{ y } \), show that \(\frac { { x }^{ m } }{ { y }^{ n } } -\frac { { y }^{ n } }{ { x }^{ m } } =2isin\left( m\alpha -n\beta \right) \)
5)
Solve the equation
2x³- 9x²+ 10x = 3

12th Standard Maths English Medium - Important 2 Mark Question Paper and Answer Key 2022 - 2023 - by Study Materials - Dec 31, 2022 - View & Read

1)
If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.
2)
If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A⁻¹.
3)
If z = x + iy, find the following in rectangular form.
\(Re\left( \frac { 1 }{ z } \right) \)
4)
If z_i= 2− i and z₂= -4+3i , find the inverse of z₁z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 2 } } \)
5)
If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

12th Standard Maths English Medium - Important 1 Mark MCQ's Question Paper and Answer Key 2022 - 2023 - by Study Materials - Dec 31, 2022 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
4)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
5)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?

12th Standard Maths Revision Model Question Paper With Answer Key - by Study Materials - Dec 31, 2022 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
4)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
5)
A zero of x³ + 64 is

12th Standard Maths English Medium - Discrete Mathematics 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z.
2)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.
3)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_e = the set of all even integers
4)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_o = the set of all odd integers
5)
Verify
(i) closure property
(ii) commutative property, and
(iii) associative property of the following operation on the given set. (a*b) = a^b;∀a, b∈N (exponentiation property)

12th Standard Maths English Medium - Discrete Mathematics 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Show that ¬( p ∧ q) ≡ ¬p V ¬q
2)
Write the truth value for each of the following statements.
(1) 3 + 5 = 8 and \(\sqrt{2}\) is an irrational number.
(2) 5 is a positive integer or a square is a rectangle.
(3) Chennai is not in Tamilnadu.
3)
In (z, *) where * is defined by a * b = a^b, prove that * is not a binary operation on z.
4)
Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.
5)
In (z, *) where * is defined as a * b = a + b + 2. Verify the commutative and associative axiom.

12th Standard Maths English Medium Maths - Discrete Mathematics 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Construct the truth table for \((p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)\)
2)
Establish the equivalence property p ➝ q ≡ ㄱp ν q
3)
Establish the equivalence property connecting the bi-conditional with conditional: p ↔️ q ≡ (p ➝ q) ∧ (q⟶ p)
4)
On Z, define \(⊗ \mathrm{by}(m * n)\) = mn + nm: ∀m, n∈Z. Is ⊗ binary on Z?
5)
Let \(*\) be defined on R by (a \(*\) b) = a + b + ab - 7. Is \(*\)binary on R? If so, find 3 \(*\)\(\left( \frac { -7 }{ 15 } \right) \).

12th Standard Maths English Medium - Discrete Mathematics 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Show that p v (~p) is a tautology.
2)
Show that p v (q ∧ r) is a contingency.
3)
In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.
4)
Let S be the set of positive rational numbers and is defined by a * b = \(\frac{ab}{2}\). Then find the identity element and the inverse of 2.
5)
Let G = {1, w, w²) where w is a complex cube root of unity. Then find the universe of w². Under usual multiplication.

12th Standard Maths English Medium - Discrete Mathematics 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
a*b = a + 3ab − 5b²; ∀a,b∈Z
2)
Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)
3)
Identify the valid statements from the following sentences.
4)
Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining'.
5)
Let A =\(\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)be any two boolean matrices of the same type. Find AvB and A\(\wedge\)B.

12th Standard Maths English Medium - Discrete Mathematics 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The binary operation * defined on a set s is said to be commutative if ______
2)
If * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is _____________
3)
The number of binary operations that can be defined on a set of 3 elements is _____________
4)
The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________
5)
Which one of the following is not a statement?

12th Standard Maths English Medium - Discrete Mathematics 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A binary operation on a set S is a function from
2)
Subtraction is not a binary operation in
3)
Which one of the following is a binary operation on N?
4)
In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?
5)
The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on

12th Standard Maths English Medium -Probability Distributions 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A pair of dice is thrown l0 times. If getting a sum 10 is success, find the probability of
(i) 10 success
(ii) No success
(iii) More than 8 success
2)
The probability function of a random variable X is f(x) \(=\mathrm{Ce}^{-|x|},-\infty<\mathrm{x}<\infty\) . Find the value of C and also find the mean and variance for the random variable.
3)
An urn contains 4 Green and 3 Red balls. Find the probability distribution of the number of red balls in 3 draws when a baII is drawn at random with replacement. Also find its mean and variance.

12th Standard Maths English Medium -Probability Distributions 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.
2)
An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.
3)
Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.
4)
A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.
5)
A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)

12th Standard Maths English Medium -Probability Distributions 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

12th Standard Maths English Medium -Probability Distributions 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.
2)
The probability density function of X is given by \(f(x)=\begin{cases} \begin{matrix} kxe^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & for\quad x\le 0 \end{matrix} \end{cases}\) Find the value of k.
3)
An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen, find the values taken by the random variable X and its number of inverse images
4)
Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.
5)
Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.

12th Standard Maths English Medium -Probability Distributions 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours;
(iii) at least 2 will not have a useful life of at : least 600 hours.

12th Standard Maths English Medium -Probability Distributions 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.
2)
Suppose two coins are tossed once. If X denotes the number of tails,
(i) write down the sample space
(ii) find the inverse image of 1
(iii) the values of the random variable and number of elements in its inverse images
3)
Compute P(X = k) for the binomial distribution, B(n, p) where
\(n=10, p=\frac{1}{5}, k=4\)

12th Standard Maths English Medium -Probability Distributions 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A die is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is _____________
2)
Var (2x ± 5) is =________
3)
If the p.d.f. \(f(x)=\{ \begin{matrix} \cfrac { x }{ 2 } ,0 then \(\\ \\ \\ E\left( { 3x }^{ 2 }-2x \right) \) =_______.
4)
The variance of a binomial distribution is________.
5)
In a binomial distribution n = 4, \(P(X=0)=\frac { 16 }{ 81 } \) then P(X = 4) is __________.

12th Standard Maths English Medium -Probability Distributions 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct
2)
A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.
3)
Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is
4)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
5)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

12th Standard Maths English Medium - Ordinary Differential Equations 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Solve : \({ e }^{ \frac { dy }{ dx } }=x+1,y(0)=5\)
2)
Solve : \(\left( 1+{ x }^{ 2 } \right) \frac { dy }{ dx } -x={ 2tan }^{ -1 }x\)
3)
Solve : \(\left( { x }^{ 2 }+{ x }^{ 2 }+x+1 \right) \frac { dy }{ dx } ={ 2x }^{ 2 }+x\)
4)
Solve : (1+y²)(1 + log x)dx + x dy = 0, given that x = 1,y = 1.
5)
In a bank principal increases at the rate of 5% per year. In how many years Rs.1000 doubled itself.

12th Standard Maths English Medium - Ordinary Differential Equations 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Express each of the following physical statements in the form of differential equation.
2)
Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
3)
Show that y = e^−x + mx + n is a solution of the differential equation e^x \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0
4)
Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)
5)
Show that y = ax + \(\frac { b }{ x } \), x ≠ 0 is a solution of the differential equation x² y" + xy' - y = 0.

12th Standard Maths English Medium - Ordinary Differential Equations 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Solve \(\frac { dy }{ dx } +\frac { { y }^{ 2 } }{ { x }^{ 2 } } =\frac { y }{ x } \)
2)
Form the differential equation for y = e^-2x [A cos 3x-B sin 3x]
3)
Solve: \(\frac{dy}{dx}=\)(4x + y + 1)²
4)
Solve: x\(\frac{dy}{dx}\)+ 2y = x²
5)
Solve: \(\frac{dy}{dx}+y=cos x\)

12th Standard Maths English Medium - Ordinary Differential Equations 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the differential equation of the family of circles passing through the points (a, 0) and (−a, 0).
2)
Solve \((1+{ x }^{ 2 })\frac { dy }{ dx } =1+{ y }^{ 2 }\)
3)
Find the particular solution of (1+ x³)dy − x² ydx = 0 satisfying the condition y(1) = 2.
4)
Solve y' = sin² (x − y + 1 ).
5)
Solve : \(\frac { dy }{ dx } =\sqrt { 4x+2y-1 } \)

12th Standard Maths English Medium - Ordinary Differential Equations 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Form the differential equation satisfied by are the straight lines in my-plane.
2)
A curve passing through the origin has its slope e^x, Find the equation of the curve.
3)
Solve: \(\frac{dy}{dx}=1+e^{x-y}\)
4)
Solve: x \(\frac{dy}{dx}=x+y\)
5)
Solve: \(\frac{dy}{dx}+y=e^{-x}\)

12th Standard Maths English Medium - Ordinary Differential Equations 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { dy }{ dx } +xy=cotx\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)
3)
For each of the following differential equations, determine its order, degree (if exists)
\({ { \left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) } }^{ 2 }+{ \left( \frac { dy }{ dx } \right) }^{ 2 }=xsin\left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) \)
4)
For each of the following differential equations, determine its order, degree (if exists)
\(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)
5)
For each of the following differential equations, determine its order, degree (if exists)
\(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

12th Standard Maths English Medium - Ordinary Differential Equations 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The order and degree of y'+(y")²= (x + t")² are _________.
2)
The differential equation corresponding to xy = c² where c is an arbitrary constant is ________.
3)
On finding the differential equation corresponding to y = e^mx where m is the arbitrary constant, then m is ________.
4)
The population p of a certain bacteria decreases at a rate proportional to the population p. The differential equation corresponding to the above statement is __________.
5)
The solution of log \(\left( \frac { dy }{ dx } \right) \) = ax + by is______.

12th Standard Maths English Medium - Ordinary Differential Equations 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively
2)
The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is
3)
The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is
4)
The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is
5)
The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is

12th Standard Maths English Medium - Applications of Integration 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the area bounded by the curve y=xe^x and y=xe^-xand the line x=1.
2)
Find the area of the region bounded by y=e^x and y=e^-x and the;line x=1.
3)
Find the area of the region bounded by a²y²=a²(a²-x²)
4)
Find the area of the region enclosed by the two circles x²+y²=1 and (x-1)²+y²=1.
5)
AOB is the positive quadrant of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) where OA=a and OB=b.Find the area between the arc AB and chord AB of the elipse.

12th Standard Maths English Medium - Applications of Integration 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\) using the Riemann sums corresponding to 5 subintervals of equal width and applying
(i) left-end rule
(ii) right-end rule
(iii) the mid-point rule.
2)
Evaluate: \(\int ^4_{-4}\) |x+3| dx.
3)
Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log_e2.
4)
Evaluate : \(\int ^\frac{\pi}{4}_{0} \frac{1}{sin x+cos x}\) dx
5)
Evaluate\(\int ^{\pi}_{0} \frac{x}{1+sin x}\) dx

12th Standard Maths English Medium - Applications of Integration 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Evaluate \(\int _{ 0 }^{ 50 }{ \left[ x-\left| x \right| \right] dx } \)
2)
Evaluate \(\int _{ -2 }^{ 3 }{ \left| 1-{ x }^{ 2 } \right| } dx\)
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 3/2 }x }{ { cos }^{ 3/2 }x+{ sin }^{ 3/2 }x } } dx\)
4)
Evaluate \(\int_{0}^{\pi / 2} x \cos x d x\)
5)
Evaluate \(\int_{0}^{1} \frac{d x}{\sqrt{1+x}-\sqrt{x}}\)

12th Standard Maths English Medium - Applications of Integration 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ xdx } \) by applying the left-end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.
2)
Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ x^2dx } \) by applying the right-end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.
3)
Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ { (2-x)dx } } \) by applying the mid-point rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.
4)
Evaluate\(\int _{ 1 }^{ 4 }{ ({ 2x }^{ 2 }+3) } \) dx, as the limit of a sum
5)
Evaluate the following integrals as the limits of sums.
\(\int _{ 0 }^{ 1 }{ (5x+4)dx } \)

12th Standard Maths English Medium - Applications of Integration 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Evaluate \(\int _{ 0 }^{ 1 }{ \left( \frac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) } \)
2)
Evaluate \(\int { \sum _{ r=0 }^{ \infty }{ \cfrac { { x }^{ r }{ 2 }^{ r } }{ r! } } dx } \)
3)
Evaluate \(\int { { e }^{ 3x }3^{ 2x }{ 5 }^{ x }dx } \)
4)
Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)
5)
Find the slope of the tangent to the curve \(y=\int _{ 0 }^{ x }{ \frac { dt }{ 1+{ t }^{ 3 } } stx=1 } \)

12th Standard Maths English Medium - Applications of Integration 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Evaluate \(\int _{ 0 }^{ 1 }{ xdx } \), as the limit of a sum.
2)
Evaluate \(\int _{ 0 }^{ 1 }{ x^3dx } \), as the limit of a sum.
3)
Evaluate :\(\int _{ 0 }^{ 1 }{ [2x] } dx\) where [⋅] is the greatest integer function
4)
Evaluate :\(\int _{ 0 }^{ \frac { \pi }{ 3 } }{ \frac { sec\ x\ tan\ x }{ 1+{ sec }^{ 2 }x } dx } \)
5)
Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)

12th Standard Maths English Medium - Applications of Integration 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The value of \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is __________
2)
The area enclosed by the curve y = \(\frac { { x }^{ 2 } }{ 2 } \) , the x - axis and the lines x = 1, x = 3 is __________
3)
The area bounded by the parabola y = x² and the line y = 2x is __________
4)
The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is __________
5)
\(\int _{ 0 }^{ \infty }{ { e }^{ -mx } } { x }^{ 7 }\) dx is __________

12th Standard Maths English Medium - Applications of Integration 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is
2)
The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is
3)
If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)
4)
The area between y² = 4x and its latus rectum is
5)
The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is

12th Standard Maths English Medium - Differentials and Partial Derivatives 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If u = tan^-1 \(\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x-y } \right) \) Prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) sin 2u.
2)
Find \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } ,\frac { { \partial }^{ 2 }f }{ { \partial y }^{ 2 } } \) at x = 2, y = 3 if f(x,y) = 2x² + 3y² - 2xy
3)
Using differential find the approximate value of cos 61; if it is given that sin 60° = 0.86603 and 1⁰ = 0.01745 radians.
4)
If V = log r and r² = x² +y² + z², then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } } \)
5)
If z = f(x - cy) + F (x + cy) where f and F are any two functions and c is a constant, show that \(\frac { { \partial }^{ 2 }z }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }z }{ \partial { y }^{ 2 } } \)

12th Standard Maths English Medium - Differentials and Partial Derivatives 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.
2)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error
3)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. Find the following in calculating the area of the circular plate:
Relative error
4)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm.find the following in calculating the area of the circular plate:
Percentage error
5)
A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:
(i) change in the volume
(ii) change in the surface area

12th Standard Maths English Medium - Differentials and Partial Derivatives 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If w= log(x²+y²) and x=rcosፀ and y=rsinፀ then, find \(\frac { \partial w }{ \partial r } and\frac { \partial w }{ \partial \theta } \)
2)
If w=xy+z and x=cot, y=sint, z=t then find \(\frac { dw }{ dt } \)
3)
Using linear approximation find \(\sqrt { 0.082 } \)
4)
Find the approximate value of \(\left( \frac { 17 }{ 81 } \right) ^{ \frac { 1 }{ 4 } }\) using linear approximation.
5)
Find the limit for the following if it exists \(\underset { (x-y)\rightarrow \left( 1,1 \right) }{ lim } \frac { { 2x }^{ 2 }-xy-{ y }^{ 2 } }{ { x }^{ 2 }-{ y }^{ 2 } } \)

12th Standard Maths English Medium - Differentials and Partial Derivatives 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the linear approximation for f(x) = \(\sqrt { 1+x } ,x\ge -1\) at x₀ = 3. Use the linear approximation to estimate f(3.2)
2)
Use linear approximation to find an approximate value of \(\sqrt { 9.2 } \) without using a calculator.
3)
Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.
4)
Let \(f(x)=\sqrt [ 3 ]{ x } \). Find the linear approximation at x = 27. Use the linear approximation to approximate \(\sqrt [ 3 ]{ 27.2 } \)
5)
Find a linear approximation for the following functions at the indicated points.
f(x) = x³ - 5x + 12, x₀ = 2

12th Standard Maths English Medium - Differentials and Partial Derivatives 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If w=e^xy,x=at²,y=2at, find \(\frac { dw }{ dt } \)
2)
If w=log(x²+y²),x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)
3)
If \(w={ e }^{ { x }^{ 2 }+{ y }^{ 2 } }\) ,x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)
4)
If w=xye^xy find \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \)
5)
Using differentials, find the approximate value of \(sin\left( \frac { 22 }{ 14 } \right) \)

12th Standard Maths English Medium - Differentials and Partial Derivatives 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Use the linear approximation to find approximate values of \({ (123) }^{ \frac { 2 }{ 3 } }\)
2)
Use the linear approximation to find approximate values of \(\sqrt [ 4 ]{ 15 } \)
3)
Use the linear approximation to find approximate values of \(\sqrt [ 3 ]{ 26 } \)
4)
Let g(x) = x² + sin x. Calculate the differential dg.
5)
Find differential dy for each of the following function \(y=\frac { { \left( 1-2x \right) }^{ 3 } }{ 3-4x } \)

12th Standard Maths English Medium - Differentials and Partial Derivatives 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If y = x⁴ - 10 and if x changes from 2 to 1.99, the approximate change in y is ________
2)
If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, the approximate error in calculating its volume is _____________
3)
If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________
4)
If u = x^y + y^x then u_x + u_y at x = y = 1 is _____________
5)
lf u = (x-y)⁴+(y-z)⁴ +(z-x)⁴ then \(\sum { \frac { \partial u }{ \partial x } } \) = _____________

12th Standard Maths English Medium - Differentials and Partial Derivatives 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is
2)
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
3)
If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to
4)
If v (x, y) = log (e^x+ e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to
5)
If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

12th Standard Maths English Medium - Application of Differential Calculus 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Gas is escaping from a spherical balloon at the rate of 900 cm³/sec. How fast is the surface area and radius of the balloon shrinking when the radius of the balloon is 30 cm?
2)
A particle moves along the curve \(y=\frac { 4 }{ 3 } { x }^{ 3 }+5\). Find the points on the curve at which y coordinate changes as fast as x-coordinates.
3)
Find the points on the curve y=2x²-2x² at which the tangent lines are parallel to the line y=3x-2.
4)
If the curves 4x=y² and 4xy=k cut at right angles show that k²=512.
5)
missle fired from ground level rises x metres vertically upwards in t seconds and \(x=100t-\frac { 25 }{ 2 } { t }^{ 2 }\). Find the
(i) initial velocity of the missile
(ii) the time when the height of the missile is maximum
(iii) the maximum height reached
(iv) the velocity which the missile strikes the ground.

12th Standard Maths English Medium - Application of Differential Calculus 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?
2)
A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.
3)
If we blow air into a balloon of spherical shape at a rate of 1000 cm³ per second. At what rate the radius of the baloon changes when the radius is 7cm? Also compute the rate at which the surface area changes.
4)
Salt is poured from a conveyer belt at a rate of 30 cubic metre per minute forming a conical pile with a circular base whose height and diameter of base are always equal. How fast is the height of the pile increasing when the pile is 10 metre high?
5)
A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A 10 kilometres to the north of P and traveling at 80 km/hr, while car B is 15 kilometres to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing?

12th Standard Maths English Medium - Application of Differential Calculus 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the equation of normal to the cure y = sin²x at \(\left( \frac { \pi }{ 3 } ,\frac { 3 }{ 4 } \right) \).
2)
Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].
3)
The ends of a rod AB which is 5 m long moves along two grooves OX, OY which at the right angles. If A moves at a constant speed of \(\frac { 1 }{ 2 } \) m/sec, what is the speed of B, when it is 4m from O?
4)
A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t² where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?
5)
The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.

12th Standard Maths English Medium - Application of Differential Calculus 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The temperature T in celsius in a long rod of length 10 m, insulated at both ends, is a function of length x given by T = x(10 − x). Prove that the rate of change of temperature at the midpoint of the rod is zero.
2)
A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) = 100 × (1− 0.1t)², 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?
3)
A particle moves so that the distance moved is according to the law s(t) = \(s(t)=\frac{t^{3}}{3}-t^{2}+3\). At what time the velocity and acceleration are zero.
4)
The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.
5)
A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.

12th Standard Maths English Medium - Application of Differential Calculus 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A man 2 m high walks at a uniform speed of 5 km/ hr away from a lamp post 6 m high. Find the rate at which the length of his shadow increases?
2)
At what point on the curve y = x² on [-2, 2] is the tangent parallel to X-axis?
3)
Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).
4)
Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.
5)
Find x if the rate of decrease of \(\frac { { x }^{ 2 } }{ 2 } -2x+5\) is twice the decrease of x.

12th Standard Maths English Medium - Application of Differential Calculus 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2
2)
A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.
3)
If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units.
4)
If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = \(\sqrt { 3 } x\) then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres.
5)
A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?

12th Standard Maths English Medium - Application of Differential Calculus 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________
2)
The angle made by any tangent to the curve y = x⁵ + 8x + 1 with the X-axis is a __________
3)
The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________
4)
The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________
5)
In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

12th Standard Maths English Medium - Application of Differential Calculus 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.
3)
The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is
4)
A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by
5)
The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is

12th Standard Maths English Medium - Applications of Vector Algebra 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.
2)
ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)
3)
If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3
4)
Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \)
5)
Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

12th Standard Maths English Medium - Applications of Vector Algebra 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
By vector method, prove that cos(α + β) = cos α cos β - sin α sin β
2)
With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)
3)
Prove by vector method that sin(α −β) = sinα cosβ −cosα sinβ
4)
If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)
5)
Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

12th Standard Maths English Medium - Applications of Vector Algebra 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)
2)
Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.
3)
If \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) then show that \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } =\overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \)
4)
Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar
5)
Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

12th Standard Maths English Medium - Applications of Vector Algebra 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A particle acted upon by constant forces \(\hat { 2j } +\hat { 5j } +\hat { 6k } \) and \(-\hat { i } -\hat { 2j } -\hat { k } \) is displaced from the point (4, −3, −2) to the point (6, 1, −3). Find the total work done by the forces.
2)
Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin
3)
Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).
4)
Find the magnitude and direction cosines of the torque of a force represented by \(\hat { 3i } +\hat { 4j } -\hat { 5k } \) about the point with position vector \(\hat { 2i } -\hat { 3j } +\hat { 4k } \) acting through a point whose position vector is \(\hat { 4i } +\hat { 2j } -\hat { 3k } \).
5)
Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \), \(\hat { i } +\hat { 2j } -\hat { k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)

12th Standard Maths English Medium - Applications of Vector Algebra 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)
2)
Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)
3)
Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.
4)
If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.
5)
Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

12th Standard Maths English Medium - Applications of Vector Algebra 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
With usual notations, in any triangle ABC, prove the following by vector method.
(i) a²= b²+ c²− 2bc cos A
(ii) b²= c²+ a²− 2ca cos B
(iii) c²= a²+ b²− 2ab cos C
2)
With usual notations, in any triangle ABC, prove the following by vector method.
(i) a = b cos C + c cos B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A
3)
A particle is acted upon by the forces \((\hat { 3i } -\hat { 2j } +\hat { 2k } )\) and \((\hat { 2i } +\hat { j } -\hat { k } )\) is displaced from the point (1, 3, -1 ) to the point (4, -1, λ). If the work done by the forces is 16 units, find the value of λ.
4)
Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.
5)
Prove by vector method that the median to the base of an isosceles triangle is perpendicular to the base.

12th Standard Maths English Medium - Applications of Vector Algebra 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________
2)
If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectors, then the vectors \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) \) is parallel to the vector ___________
3)
The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +\overset { \wedge }{ 4k } \) is ____________
4)
If \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are any three vectors, then \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \) if and only if __________
5)
The volume of the parallelepiped whose sides are given by \(\overset { \rightarrow }{ OA } =2\overset { \wedge }{ i } -3\overset { \wedge }{ j } \), \(\overset { \rightarrow }{ OB } =\overset { \wedge }{ i } +\overset { \wedge }{ j } -\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ OC } =3\overset { \wedge }{ i } -\overset { \wedge }{ k } \) is _____________

12th Standard Maths English Medium - Applications of Vector Algebra 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
3)
If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is
4)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to
5)
If \([\vec{a}, \vec{b}, \vec{c}]=1\), then the value of \(\frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{c} \times \vec{a})}{(\vec{a} \times \vec{b}) \cdot \vec{c}}+\frac{\vec{c} \cdot(\vec{a} \times \vec{b})}{(\vec{c} \times \vec{b}) \cdot \vec{a}}\) is

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the equation of the tangent at t = 1 to the parabola y² = 12x
2)
Find the equations of the two tangents that can be drawn from the point (5, 2) to the ellipse 2x² +7y² = 14.
3)
Find the vertex, focus, directrix, axis and latus rectum of the parabola \(y^{2}-4 x-4 y=0\)
4)
Find the equation of the ellipse given that the centre is (4, -1), focus is (1, -1) and passing through (8, 0).
5)
Find the equation of a point which moves so that the sum of its distances ftom (- 4, 0) and (4, 0) is 10.

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the equation of the circle passing through the points (1, 1 ), (2, -1 ) and (3, 2) .
2)
A road bridge over an irrigation canal have two semi circular vents each with a span of 20m and the supporting pillars of width 2m. Use Figure to write the equations that represents the semi-verticular vents
3)
Find the equation of the circle through the points (1, 0),(-1, 0) , and (0, 1)
4)
Determine whether the points (-2, 1), (0, 0) and (-4, -3) lie outside, on or inside the circle x²+y²−5x+2y−5 = 0 .
5)
If the equation 3x²+(3−p)xy+qy²−2px = 8pq represents a circle, find p and q. Also determine the centre and radius of the circle

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13.
2)
For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.
3)
Find the value of c if y = x + c is a tangent to the hyperbola 9x² - 16y² = 144.
4)
Show that the line x + y + 1 = 0 touches the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 15 } \) = 1 and find the co-ordinates of the point of contact
5)
Find the equation of the circle, which is concentric with the circle \(x^{2}+y^{2}-4 x-6 y-9=0\) and passing through the point (- 4, - 5).

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
A line 3x+4y+10 = 0 cuts a chord of length 6 units on a circle with centre of the circle (2,1). Find the equation of the circle in general form.
2)
Find the equations of the tangent and normal to the circle x²+ y²= 25 at P(-3, 4).
3)
Find the length of Latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\)
4)
Find the equation of the hyperbola with vertices (0, ±4) and foci(0, ±6).
5)
Find the equation of the ellipse in each of the cases given below:
length of latus rectum 4, distance between foci 4 \( \sqrt{ 2}\) , centre (0, 0) and major axis as y - axis.

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.
2)
For the ellipse x² + 3y² = a², find the length of major and minor axis.
3)
Find the eccentricity of the ellipse with foci on x-axis if its latus rectum be equal to one half of its major axis.
4)
Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.
5)
Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
Find the general equation of a circle with centre (-3, -4) and radius 3 units.
2)
Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x²+ y²= 16 as diameter.
3)
Determine whether x + y − 1 = 0 is the equation of a diameter of the circle x²+ y²− 6x + 4y + c = 0 for all possible values of c .
4)
Find the general equation of the circle whose diameter is the line segment joining the points (−4, −2)and (1, 1) is x²+y²+5x+3y+6=0
5)
Examine the position of the point (2, 3) with respect to the circle x²+ y²− 6x − 8y + 12 = 0.

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________
2)
The eccentricity of the ellipse 9x²+ 5y² - 30y = 0 is __________
3)
The length of the latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is __________
4)
If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is __________
5)
In an ellipse, the distance between its foci is 6 and its minor axis is 8, then e is ________

12th Standard Maths English Medium - Two Dimensional Analytical Geometry-II 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 02, 2022 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
3)
The circle x²+ y²= 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if
4)
The length of the diameter of the circle which touches the x - axis at the point (1, 0) and passes through the point (2, 3).
5)
The radius of the circle 3x²+ by²+ 4bx − 6by + b² = 0 is

12th Standard Maths English Medium - Inverse Trigonometric Functions 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)
2)
Find the domain of the following functions
(i) f(x) = sin^-1(2x - 3)
(ii) f(x) = sin^-1x + cos x
3)
Write the function \(f(x)=\tan ^{-1} \sqrt{\frac{a-x}{a+x}}-a<x<a \) in the simplest form
4)
Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\)
5)
If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x²= sin 2a

12th Standard Maths English Medium - Inverse Trigonometric Functions 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1\((\frac{1-|x|}{4})\)
2)
Find the value of tan⁻¹(−1 ) + cos^-1\((\frac{1}{2})+sin^-1(-\frac{1}{2})\)
3)
If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)
4)
Prove that tan^-1 x + tan^-1 z = tan^-1\(\left[ \frac { x+y+z-xyz }{ 1-xy-yz-zx } \right] \)
5)
If tan^-1 x + tan^-1y + tan^-1z = \(\pi\), show that x + y + z = xyz

12th Standard Maths English Medium - Inverse Trigonometric Functions 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Prove that \({ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) ={ tan }^{ -1 }\left( \frac { 27 }{ 11 } \right) \)
2)
Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)
3)
Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)
4)
Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)
5)
If \(sin\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.

12th Standard Maths English Medium - Inverse Trigonometric Functions 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the domain of sin⁻¹(2−3x²)
2)
Find all the values of x such that -10\(\pi\)\(\le x\le\)10\(\pi\) and sin x = 0
3)
Find the domain of the following
\(f\left( x \right) { =sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right) \)
4)
Find the value of sin^-1\(\left( sin\frac { 5\pi }{ 9 } cos\frac { \pi }{ 9 } +cos\frac { 5\pi }{ 9 } sin\frac { \pi }{ 9 } \right) \).
5)
Find the domain of cos^-1\((\frac{2+sinx}{3})\)

12th Standard Maths English Medium - Inverse Trigonometric Functions 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find all the values of x such that
-3\(\pi\)\(\le x\le\)-3\(\pi\) and sin x = -1
2)
Find the principal value of \({ tan }^{ -1 }\left( \frac { -1 }{ \sqrt { 3 } } \right) \)
3)
Find the principal value of sin^-1(-1).
4)
Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)
5)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

12th Standard Maths English Medium - Inverse Trigonometric Functions 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the principal value of sin^-1\(\left( -\frac { 1 }{ 2 } \right) \)(in radians and degrees).
2)
Find the principal value of sin^-1(2), if it exists.
3)
Find the principal value of
\({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)
4)
Find the period and amplitude of y = sin 7x
5)
Sketch the graph of y = sin\((\frac{1}{3}x)\) for 0\(\le x <6\pi\).

12th Standard Maths English Medium - Inverse Trigonometric Functions 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________
2)
The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi is ___________
3)
If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________
4)
\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________
5)
If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________

12th Standard Maths English Medium - Inverse Trigonometric Functions 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to
3)
\(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}+\sec ^{-1} \frac{5}{3}-\operatorname{cosec}^{-1} \frac{13}{12}\) is equal to
4)
If sin⁻¹x = 2sin⁻¹\(\alpha\) has a solution, then
5)
\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for

12th Standard Maths English Medium - Theory of Equations 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
If the sum of the roots of the quadratic equation ax²+ bx + c = 0 (abc ≠ 0) is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \) are H.P.
2)
If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc
3)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
4)
If the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x² + ax + bc = 0.
5)
Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

12th Standard Maths English Medium - Theory of Equations 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Solve the equation x³− 9x²+14x + 24 = 0 if it is given that two of its roots are in the ratio 3:2.
2)
If α, β, and γ are the roots of the polynomial equation ax³+ bx²+ cx + d = 0, find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.
3)
If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.
4)
If the equations x²+ px + q = 0 and x²+ p'x + q' = 0 have a common root, show that it must be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).
5)
A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

12th Standard Maths English Medium - Theory of Equations 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.
2)
If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².
3)
If α, β, and γ are the roots of the equation x³+ px²+ qx + r = 0, find the value of \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.
4)
If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.
5)
Solve the equation 3x³- 16x²+ 23x - 6 = 0 if the product of two roots is 1.

12th Standard Maths English Medium - Theory of Equations 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the number of real solutions of sin (e^x) -5^x + 5^-x
2)
Find the number of positive integral solutions of (pairs of positive integers satisfying) x² - y² = 353702.
3)
Solve: 2x+2^x-1+2^x-2= 7x+7^x-1+7^x-2
4)
Solve: (x-1)⁴+(x-5)⁴= 82
5)
Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

12th Standard Maths English Medium - Theory of Equations 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Formulate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.
2)
Construct a cubic equation with roots 2, −2, and 4.
3)
If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²
4)
Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.
5)
Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.

12th Standard Maths English Medium - Theory of Equations 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Construct a cubic equation with roots 1, 2 and 3
2)
If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ
3)
If p is real, discuss the nature of the roots of the equation 4x²+ 4px + p + 2 = 0 in terms of p.
4)
If α, β, γ and \(\delta\) are the roots of the polynomial equation 2x⁴+ 5x³− 7x²+ 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).
5)
Find the monic polynomial equation of minimum degree with real coefficients having 2 -\(\sqrt{3}\)i as a root.

12th Standard Maths English Medium - Theory of Equations 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________
2)
The quadratic equation whose roots are ∝ and β is ___________
3)
If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots
4)
If x is real and \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then ________
5)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________

12th Standard Maths English Medium - Theory of Equations 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
A zero of x³ + 64 is
2)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
3)
A polynomial equation in x of degree n always has
4)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
5)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?

12th Standard Maths English Medium - Complex Numbers 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Prove that the values of \(\sqrt [ 4 ]{ -1 } arr\ \pm \frac { 1 }{ \sqrt { 2 } } \left( 1\pm i \right) \). Let z = (-1)
2)
If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) = 64
3)
Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1
4)
Verify that 2 arg(-1) ≠ arg(-1)²
5)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

12th Standard Maths English Medium - Complex Numbers 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.
2)
If z₁= 2 + 5i, z₂= -3 - 4i, and z₃= 1 + i, find the additive and multiplicate inverse of z₁, z₂ and z₃
3)
Show that \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ 1+2i } \right) ^{ 15 }\) is purely imaginary.
4)
Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }-\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is purely imaginary
5)
Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r

12th Standard Maths English Medium - Complex Numbers 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.
2)
Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.
3)
If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).
4)
Find the value of \(\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1\)
5)
Show that \(i^{n+100}+i^{n+50}+i^{n+48}+i^{n+46}=0, \forall n \in N\)

12th Standard Maths English Medium - Complex Numbers 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3 and x+(−1+2i)y+1+i are equal
2)
If z₁= 1 - 3i, z₂= - 4i, and z₃ = 5 , show that (z₁+ z₂) + z₃= z₁+ (z₂+ z₃)
3)
If z₁= 3, z₂= -7i, and z₃= 5 + 4i, show that z₁(z₂+ z₃) = z₁z₂+ z₁z₃
4)
Write \(\frac { 3+4i }{ 5-12i } \) in the x + iy form, hence find its real and imaginary parts.
5)
Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form

12th Standard Maths English Medium - Complex Numbers 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?
2)
Find Re (z) and im (z) if z = 5i¹¹ + 7i³
3)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
4)
If z₁ and z₂ are 1-i, -2+4i then find Im\(\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { z }_{ 1 } } } \right) \).
5)
If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0

12th Standard Maths English Medium - Complex Numbers 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
Simplify the following i⁷
2)
Simplify the following
i¹⁹⁴⁷+ i¹⁹⁵⁰
3)
Evaluate the following if z = 5−2i and w = −1+3i
z + w
4)
Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, iz , and z+iz
5)
Write the following in the rectangular form:
\(\overline { \left( 5+9i \right) +\left( 2-4i \right) } \)

12th Standard Maths English Medium - Complex Numbers 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Sep 01, 2022 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
3)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
4)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
5)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to

12th Standard Maths English Medium Application of Matrices and Determinants 5 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 24, 2022 - View & Read

1)
Using determinants; find the quadratic defined by f(x) = ax² + bx + c, if f(1) = 0, f(2) = -2 and f(3) = -6.
2)
Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2
3)
The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.
4)
For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.
5)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.

12th Standard Maths English Medium Application of Matrices and Determinants 5 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 24, 2022 - View & Read

1)
Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).
2)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
3)
If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.
4)
A = \(\left[ \begin{matrix} 1 & \tan { x } \\ -\tan { x } & 1 \end{matrix} \right] \), show that A^TA^-1 = \(\left[ \begin{matrix} \cos { 2x } & -\sin { 2x } \\ \sin { 2x } & \cos { 2x } \end{matrix} \right] \)
5)
Find the matrix A for which A\(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] =\left[ \begin{matrix} 14 & 7 \\ 7 & 7 \end{matrix} \right] \).

12th Standard Maths English Medium Application of Matrices and Determinants 3 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 24, 2022 - View & Read

1)
Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.
2)
For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?
3)
Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.
4)
Verify (AB)^-1 = B^-1 A^-1 for A =\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).
5)
Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

12th Standard Maths English Medium Application of Matrices and Determinants 3 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 24, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A) = (adj A)A = |A| I₃.
2)
Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).
3)
Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).
4)
If adj A = \(\left[ \begin{matrix} -1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), find A⁻¹.
5)
Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

12th Standard Maths English Medium Application of Matrices and Determinants 2 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 24, 2022 - View & Read

1)
For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.
2)
For the matrix A, if A³ = I, then find A^-1.
3)
If A is a square matrix such that A³ = I, then prove that A is non-singular.
4)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
5)
Find the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).

12th Standard Maths English Medium Application of Matrices and Determinants 2 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 24, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
3)
If A is symmetric, prove that then adj A is also symmetric.
4)
Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.
5)
Find the adjoint of the following:
\(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

12th Standard Maths English Medium Application of Matrices and Determinants 1 Mark Creative Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 23, 2022 - View & Read

1)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
2)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
3)
If A^T is the transpose of a square matrix A, then ___________
4)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
5)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

12th Standard Maths English Medium Application of Matrices and Determinants 1 Mark Book Back Question Paper and Answer Key 2022 - 2023 - by Study Materials - Aug 23, 2022 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
4)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
5)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =

12th Standard English Medium Maths Reduced Syllabus Annual Exam Model Question Paper With Answer Key - 2021 - by Question Bank Software - Feb 27, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
2)
Which of the following is not an elementary transformation?
3)
\(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =__________
4)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is
5)
If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots

12th Standard English Medium Maths Reduced Syllabus Annual Exam Model Question Paper - 2021 - by Question Bank Software - Feb 27, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
2)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
3)
If A^T is the transpose of a square matrix A, then ___________
4)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
5)
The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is

12th Standard English Medium Maths Reduced Syllabus Public Exam Model Question Paper With Answer Key - 2021 - by Question Bank Software - Feb 27, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
3)
The solution of the equation |z| - z = 1 + 2i is
4)
If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to
5)
If sin⁻¹x = 2sin⁻¹\(\alpha\) has a solution, then

12th Standard English Medium Maths Reduced Syllabus Public Exam Model Question Paper - 2021 - by Question Bank Software - Feb 27, 2021 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
3)
The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) = 1 lies on _________
4)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
5)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

12th Standard English Medium Maths Reduced Syllabus Creative Three Mark Questions with Answer key - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.
2)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
3)
If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.
4)
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.
5)
Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

12th Standard English Medium Maths Reduced Syllabus Creative Two Mark Questions with Answer key - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
For the matrix A, if A³ = I, then find A^-1.
2)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
3)
Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
4)
Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.
5)
Find Re (z) and im (z) if z = 5i¹¹ + 7i³

12th Standard English Medium Maths Reduced Syllabus Creative one Mark Questions with Answer key - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________
2)
If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--
3)
If \(\rho\) (A) = r then which of the following is correct?
4)
If \(\rho\) (A) ≠ \(\rho\) ([AIB]), then the system is _____________
5)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

12th Standard English Medium Maths Reduced Syllabus Five Mark Important Questions With Answer Key - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.
2)
If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)
3)
Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)
4)
Find the equation of the ellipse whose eccentricity is \(\frac { 1 }{ 2 } \), one of the foci is(2, 3) and a directrix is x = 7. Also find the length of the major and minor axes of the ellipse.
5)
Certain telescopes contain both parabolic mirror and a hyperbolic mirror. In the telescope shown in figure the parabola and hyperbola share focus F₁ which is 14m above the vertex of the parabola. The hyperbola’s second focus F₂ is 2m above the parabola’s vertex. The vertex of the hyperbolic mirror is 1m below F₁. Position a coordinate system with the origin at the centre of the hyperbola and with the foci on the y-axis. Then find the equation of the hyperbola.

12th Standard English Medium Maths Reduced Syllabus Five Mark Important Questions - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Solve the following system of equations, using matrix inversion method:
2x₁ + 3x₂ + 3x₃ = 5, x₁ - 2x₂ + x₃ = -4, 3x₁ - x₂ - 2x₃ = 3.
2)
If A = \(\left[ \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right] \), find the products AB and BA and hence solve the system of equations x - y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.
3)
Test for consistency and if possible, solve the following systems of equations by rank method.
2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4
4)
Find the value of k for which the equations
kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution
5)
Solve the equation z³+ 27 = 0

12th Standard English Medium Maths Reduced Syllabus Three Mark Important Questions With Answer Key- 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.
2)
Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)
3)
If z_i= 2− i and z₂= -4+3i , find the inverse of z₁z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 2 } } \)
4)
If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)
5)
Show that \(\left( \frac { 19-7i }{ 9+i } \right) ^{ 12 }+\left( \frac { 20-5i }{ 7-6i } \right) ^{ 12 }\) is real

12th Standard English Medium Maths Reduced Syllabus Three Mark Important Questions - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Given A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right] \) and C = \(\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right] \), find a matrix X such that A X B = C.
2)
Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ 5 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} -1 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 4 \\ 11 \end{matrix} \end{matrix} \right] \)
3)
A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution? (Use Cramer’s rule to solve the problem).
4)
Find the adjoint of the following:
\(\frac { 1 }{ 3 } \left[ \begin{matrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{matrix} \right] \)
5)
The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

12th Standard English Medium Maths Reduced Syllabus Two Mark Important Questions with Answer key - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).
2)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)
3)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)
4)
Find the following \(\left| \frac { 2+i }{ -1+2i } \right| \)
5)
Which one of the points i, −2 + i, and 3 is farthest from the origin?

12th Standard English medium Maths Reduced Syllabus Two Mark Important Questions - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)
2)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right] \)
3)
Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.
4)
Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).
5)
If z₁= 1 - 3i, z₂= - 4i, and z₃ = 5 , show that (z₁+ z₂) + z₃= z₁+ (z₂+ z₃)

12th Standard English Medium Maths Reduced syllabus One Mark Important Questions With Answer Key - 2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
2)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
3)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =
4)
If x^ay^b = e^m, x^cy^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,
5)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

12th Standard English Medium Maths Reduced syllabus One Mark Important Questions -2021(Public Exam ) - by Question Bank Software - Feb 27, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \), then k =
3)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I
4)
The principal argument of (sin 40°+i cos 40°)⁵ is
5)
The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) = 1 lies on _________

12th Standard Maths English Medium Discrete Mathematics Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on
2)
In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?
3)
Which one of the following statements has the truth value T?
4)
In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are
5)
The proposition p ∧ (¬p ∨ q) is

12th Standard Maths English Medium Discrete Mathematics Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
A binary operation on a set S is a function from
2)
Which one of the following is a binary operation on N?
3)
Which one of the following statements has the truth value T?
4)
If a compound statement involves 3 simple statements, then the number of rows in the truth table is
5)
The truth table for (p ∧ q) ∨ ¬q is given below

p q (p ∧ q) ∨ (¬q)

T T (a)

T F (b)

F T (c)

F F (d)

Which one of the following is true?

12th Standard Maths English Medium Probability Distributions Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
2)
On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is
3)
If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).
4)
If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is
5)
The random variable X has the probability density function
\(f(x)=\left\{\begin{array}{lr} a x+b & 0<x<1 \\ 0 & \text { otherwise } \end{array}\right.\) and \(E(X)=\frac { 7 }{ 12 } \), then a and b are respectively

12th Standard Maths English Medium Probability Distributions Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.
2)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
3)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
4)
Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are
5)
Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E(X) is

12th Standard Maths English Medium Ordinary Differential Equations Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is
2)
The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)
3)
The degree of the differential equation \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is
4)
If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When
5)
The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

12th Standard Maths English Medium Ordinary Differential Equations Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively
2)
The solution of \(\frac{d y}{d x}+p(x) y=0\) is
3)
The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is
4)
The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)
5)
The degree of the differential equation \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is

12th Standard Maths English Medium Applications of Integration Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is
2)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)
3)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
4)
The volume of solid of revolution of the region bounded by y²= x(a − x) about x-axis is
5)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is

12th Standard Maths English Medium Applications of Integration Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is
2)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
3)
The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is
4)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is
5)
For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

12th Standard Maths English Medium Differentials and Partial Derivatives Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
2)
The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is
3)
If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to
4)
If \(f(x)=\frac{x}{x+1}\), then its differential is given by
5)
If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

12th Standard Maths English Medium Differentials and Partial Derivatives Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to
2)
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
3)
The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is
4)
The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is
5)
If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

12th Standard Maths English Medium Applications Of Differential Calculus Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by
2)
The tangent to the curve y² - xy + 9 = 0 is vertical when
3)
What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)
4)
The function sin⁴ x + cos⁴x is increasing in the interval
5)
The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

12th Standard Maths English Medium Applications Of Differential Calculus Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
3)
The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is
4)
Angle between y² = x and x²= y at the origin is
5)
What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)

12th Standard Maths English Medium Applications of Vector Algebra Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
2)
If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is
3)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to
4)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is
5)
If the volume of the parallelepiped with \(\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } \) as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \((\vec { a } \times \vec { b } )\times (\vec { b } \times \vec { c } ),(\vec { b } \times \vec { c } )\times (\vec { c } \times \vec { a } )\) and \((\vec { c } \times \vec { a } )\times (\vec { a } \times \vec { b } )\)as coterminous edges is,

12th Standard Maths English Medium Applications of Vector Algebra Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is
3)
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
4)
Consider the vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d} \) such that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\) = \(\vec { 0 } \) Let \({ P }_{ 1 }\) and \({ P }_{ 2 }\) be the planes determined by the pairs of vectors \(\vec { a } ,\vec { b } \) and \(\vec { c } ,\vec { d } \) respectively. Then the angle between \({ P }_{ 1 }\) and \({ P }_{ 2 }\) is
5)
If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are

12th Standard Maths English Medium Two Dimensional Analytical Geometry-II Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
3)
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)
4)
If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)²+ (y + 2)²= r² , then the value of r² is
5)
If x + y = k is a normal to the parabola y² = 12x, then the value of k is

12th Standard Maths English Medium Two Dimensional Analytical Geometry-II Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
2)
The circle x²+ y²= 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if
3)
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)
4)
Tangents are drawn to the hyperbola \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is
5)
An ellipse has OB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

12th Standard Maths English Medium Inverse Trigonometric Functions Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is
3)
If \(\sin ^{-1} x+\cot ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}\), then x is equal to
4)
The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________
5)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________

12th Standard Maths English Medium Inverse Trigonometric Functions Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is
2)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
3)
If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to
4)
The equation \(\tan ^{-1} x-\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)has
5)
If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________

12th Standard Maths English Medium Theory of Equations Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
A zero of x³ + 64 is
2)
A polynomial equation in x of degree n always has
3)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
4)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
5)
The polynomial x³+ 2x + 3 has

12th Standard Maths English Medium Theory of Equations Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
A zero of x³ + 64 is
2)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
3)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
4)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
5)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

12th Standard Maths English Medium Complex Numbers Reduced Syllabus Important Questions With Answer Key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
2)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
3)
If \(\alpha \) and \(\beta \) are the roots of x²+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is
4)
The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is
5)
The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________

12th Standard Maths English Medium Complex Numbers Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
3)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
4)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
5)
If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is

12th Standard Maths English Medium Application of Matrices and Determinants Reduced Syllabus Important Questions With Answer key 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
2)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
3)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
4)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
5)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

12th Standard Maths English Medium Application of Matrices and Determinants Reduced Syllabus Important Questions 2021 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
3)
If x^ay^b = e^m, x^cy^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,
4)
Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is
5)
If A is a square matrix of order n, then |adj A| = ______________

12th Standard Maths English Medium Reduced Syllabus Model Question paper with answer key - 2021 Part - 2 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
2)
Cramer's rule is applicable only when ______
3)
In a homogeneous system if \(\rho\) (A) =\(\rho\) ([A|0]) < the number of unknouns then the system has ________
4)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
5)
If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

12th Standard Maths English Medium Reduced Syllabus Model Question paper with answer key - 2021 Part - 1 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
2)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =
3)
If A^TA⁻¹ is symmetric, then A² =
4)
If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is
5)
Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

12th Standard Maths English Medium Reduced Syllabus Model Question paper - 2021 Part - 1 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
2)
If A^TA⁻¹ is symmetric, then A² =
3)
If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________
4)
The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______
5)
Equation of tangent at (-4, -4) on x² = -4y is _____________

12th Standard Maths English Medium Reduced Syllabus Model Question paper - 2021 Part - 2 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
2)
The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is
3)
If x^ay^b = e^m, x^cy^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,
4)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
5)
If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is

12th Standard Maths English Medium Reduced Syllabus Important Questions with Answer key - 2021 Part - 2 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If A^TA⁻¹ is symmetric, then A² =
3)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I
4)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
5)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

12th Standard Maths English Medium Reduced Syllabus Important Questions with Answer key - 2021 Part - 1 - by Question Bank Software - Feb 02, 2021 - View & Read

1)
Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is
2)
Which of the following is not an elementary transformation?
3)
If (1, -3) is the centre of the circle x²+ y²+ ax + by + 9 = 0 its radius is _________
4)
The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is ________
5)
The two planes 3x + 3y - 3z - 1 = 0 and x + y - z + 5 = 0 are _____________

12th Standard Maths English Medium Reduced Syllabus Important Questions - 2021 Part - 2 - by Question Bank Software - Jan 29, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
3)
The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is
4)
z₁, z₂ and z₃ are complex number such that z₁+ z₂+ z₃= 0 and |z₁| = |z₂| = |z₃| = 1 then z₁²+ z₂²+ z₃³ is
5)
If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

12th Standard Maths English Medium Reduced Syllabus Important Questions - 2021 Part - 1 - by Question Bank Software - Jan 29, 2021 - View & Read

1)
The least value of a when f f(x) = x²+ ax + 1 is increasing on (1, 2) is __________
2)
The curve y = e^x is ________
3)
If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............
4)
If u = y sin x then \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ..........
5)
The value of \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is __________

Seventh Standard Social Science CIV - Market and Consumer Protection English Medium Free Online Test 1 Mark Questions with Answer Key 2020 - 2021 - by NALLAMAYAN T - Dec 14, 2020 - View & Read

1)
In which case a consumer cannot complain against the manufacturer for a defective product?
2)
Consumer’s face various problems from the producer’s end due to _______.
3)
Consumers must be provided with adequate information about a product to make _______.
4)
The system of consumer courts at the national, state, and district levels, looking into consumers grievances against unfair trade practices of businessmen and providing necessary compensation, is called_______.
5)
Mixing other extraneous material of inferior quality with a superior quality material is called _______.

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
3)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
4)
A zero of x³ + 64 is
5)
For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
2)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
3)
If A is a non-singular matrix then IA^-1| = ______
4)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
5)
.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Two - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
2)
The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is
3)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
4)
In a homogeneous system if \(\rho\) (A) =\(\rho\) ([A|0]) < the number of unknouns then the system has ________
5)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is

12th Standard Maths English Medium Free Online Test Book Back One Mark Questions - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
3)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
4)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
5)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

12th Standard Maths English Medium Free Online Test Book Back 1 Mark Questions with Answer Key - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
2)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I
3)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
4)
\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to
5)
The centre of the circle inscribed in a square formed by the lines x²− 8x − 12 = 0 and y²− 14y + 45 = 0 is

12th Standard Maths English Medium Free Online Test Book Back One Mark Questions - Part Two - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
3)
The polynomial x³+ 2x + 3 has
4)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
5)
The length of the diameter of the circle which touches the x - axis at the point (1, 0) and passes through the point (2, 3).

12th Standard Maths English Medium Free Online Test Book Back 1 Mark Questions with Answer Key - Part Two - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If \(\sin ^{-1} x+\cot ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}\), then x is equal to
2)
The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is
3)
The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0
4)
The maximum value of the function \(x^{2} e^{-2 x}, x>0\) is
5)
If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

12th Standard Maths English Medium Free Online Test Book Back One Mark Questions - Part Three - by Question Bank Software - Nov 05, 2020 - View & Read

1)
Let C be the circle with centre at(1, 1) and radius = 1. If T is the circle centered at (0, y) passing through the origin and touching the circle C externally, then the radius of T is equal to
2)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to
3)
If the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k = 0, then the values of k are
4)
One of the closest points on the curve x² - y² = 4 to the point (6, 0) is
5)
If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

12th Standard Maths English Medium Free Online Test Creative 1 Mark Questions - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
2)
.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________
3)
If x_r= \(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x₁, x_2,x₃ ... x_∞ is _________
4)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
5)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________

12th Standard Maths English Medium Free Online Test Book Back 1 Mark Questions with Answer Key - Part Three - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
2)
If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is
3)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
4)
If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to
5)
If P(x, y) be any point on 16x²+ 25y²= 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁ + PF₂ is

12th Standard Maths English Medium Free Online Test Creative One Mark Questions with Answer Key - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
2)
If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA⁻¹ = A, then λ is
3)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
4)
If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is
5)
The equation of the normal to the circle x²+ y²− 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is

12th Standard Maths English Medium Free Online Test Creative 1 Mark Questions - Part Two - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________
2)
If a = 3 + i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and - az are ___________
3)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
4)
If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________
5)
y²- 2x - 2y + 5 = 0 is a _________

12th Standard Maths English Medium Free Online Test Creative One Mark Questions with Answer Key - Part Two - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is inconsistent then k is ___________
2)
If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________
3)
(1+i)³ = ______
4)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
5)
If u = y^x then \(\frac { \partial u }{ \partial y } \) = ............

12th Standard Maths English Medium Free Online Test Creative 1 Mark Questions - Part Three - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________
2)
.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________
3)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
4)
If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________
5)
The director circle of the ellipse \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 5 } =1\) is ____________

12th Standard Maths English Medium Free Online Test Creative One Mark Questions with Answer Key - Part Three - by Question Bank Software - Nov 05, 2020 - View & Read

1)
Which of the following is not an elementary transformation?
2)
If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________
3)
If z₁, z₂, z₃ are the vertices of a parallelogram, then the fourth vertex z₄ opposite to z₂ is _____
4)
If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________
5)
If \({ tan }^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \frac { x-1 }{ x } \right) ={ tan }^{ -1 }\left( -7 \right) \) then x is ___________

12th Standard Maths English Medium Free Online Test Creative 1 Mark Questions - Part Four - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________
2)
If a = 3 + i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and - az are ___________
3)
If x is real and \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then ________
4)
The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi is ___________
5)
Equation of tangent at (-4, -4) on x² = -4y is _____________

12th Standard Maths English Medium Free Online Test Creative One Mark Questions with Answer Key - Part Four - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A^TA⁻¹ is symmetric, then A² =
2)
Every homogeneous system ______
3)
If A is a non-singular matrix then IA^-1| = ______
4)
If x + iy = \(\frac { 3+5i }{ 7-6i } \), they y = ___________
5)
If x + y = 8, then the maximum value of xy is _________

12th Standard Maths English Medium Free Online Test Creative 1 Mark Questions - Part Five - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--
2)
If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________
3)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
4)
\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________
5)
The eccentricity of the ellipse 9x²+ 5y² - 30y = 0 is __________

12th Standard Maths English Medium Free Online Test Creative One Mark Questions with Answer Key - Part Five - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
2)
If \(\rho\) (A) = r then which of the following is correct?
3)
If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________
4)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
5)
Equation of tangent at (-4, -4) on x² = -4y is _____________

12th Standard Maths English Medium Free Online Test Volume 1 One Mark Questions 2020 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
2)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________
3)
In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______
4)
If A = [2 0 1] then the rank of AA^T is ______
5)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

12th Standard Maths English Medium Free Online Test Volume 1 One Mark Questions with Answer Key 2020 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________
2)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________
3)
If A is a non-singular matrix then IA^-1| = ______
4)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
5)
If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to

12th Standard Maths English Medium Free Online Test Volume 2 One Mark Questions 2020 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
2)
The equation of the tangent to the curve y = x²-4x+2 at (4, 2) is __________
3)
In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________
4)
If the curves y = 2e^x and y = ae^-x intersect orthogonally, then a = _________
5)
If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

12th Standard Maths English Medium Free Online Test Volume 2 One Mark Questions with Answer Key 2020 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
2)
What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)
3)
The equation of the tangent to the curve y = x²-4x+2 at (4, 2) is __________
4)
If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to
5)
If log_e4 = 1.3868, then log_e4.01 = _____________

12th Standard Maths Application of Matrices and Determinants English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
3)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
4)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
5)
If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

12th Standard Maths Application of Matrices and Determinants English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
2)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
3)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =
4)
If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA⁻¹ = A, then λ is
5)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

12th Standard Maths Complex Numbers English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
3)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
4)
z₁, z₂ and z₃ are complex number such that z₁+ z₂+ z₃= 0 and |z₁| = |z₂| = |z₃| = 1 then z₁²+ z₂²+ z₃³ is
5)
The principal argument of (sin 40°+i cos 40°)⁵ is

12th Standard Maths Complex Numbers English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
2)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
3)
The solution of the equation |z| - z = 1 + 2i is
4)
If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is
5)
The value of \(\left( \cfrac { 1+\sqrt { 3 } i}{ 1-\sqrt { 3}i } \right) ^{ 10 }\) is

12th Standard Maths Theory of Equations English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
A zero of x³ + 64 is
2)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
3)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
4)
The polynomial x³+ 2x + 3 has
5)
If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots

12th Standard Maths Theory of Equations English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
2)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
3)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
4)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is
5)
The quadratic equation whose roots are ∝ and β is ___________

12th Standard Maths Inverse Trigonometric Functions English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is
3)
If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is
4)
\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation
5)
If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is

12th Standard Maths Inverse Trigonometric Functions English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to
2)
If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is
3)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
4)
\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to
5)
sin^-1(2cos²x-1)+cos^-1(1-2sin²x)=

12th Standard Maths Two Dimensional Analytical Geometry-II English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The radius of the circle 3x²+ by²+ 4bx − 6by + b² = 0 is
3)
The equation of the normal to the circle x²+ y²− 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is
4)
If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)²+ (y + 2)²= r² , then the value of r² is
5)
The equation of the circle passing through the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) having centre at (0, 3) is

12th Standard Maths Two Dimensional Analytical Geometry-II English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
2)
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)
3)
Tangents are drawn to the hyperbola \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is
4)
Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is
5)
The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x²− 9y²= 144 are the roots of x²− (a + b)x − 4 = 0, then the value of (a+b) is

12th Standard Maths Applications of Vector Algebra English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
3)
If the volume of the parallelepiped with \(\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } \) as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \((\vec { a } \times \vec { b } )\times (\vec { b } \times \vec { c } ),(\vec { b } \times \vec { c } )\times (\vec { c } \times \vec { a } )\) and \((\vec { c } \times \vec { a } )\times (\vec { a } \times \vec { b } )\)as coterminous edges is,
4)
If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are
5)
The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

12th Standard Maths Differentials and Partial Derivatives English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If v (x, y) = log (e^x+ e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to
2)
If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to
3)
If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to
4)
If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, the approximate error in calculating its volume is _____________
5)
If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________

12th Standard Maths Applications of Vector Algebra English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
2)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is
3)
If the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -5 }= \frac {z+2 }{ 2 } \) lies in the plane x + 3y - αz + β = 0, then (α, β) is
4)
The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { j } -\hat { k) } \) represents a straight line passing through the points
5)
Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________

12th Standard Maths Application of Differential Calculus English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is
3)
The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is
4)
One of the closest points on the curve x² - y² = 4 to the point (6, 0) is
5)
If a particle moves in a straight line according to s = t³-6t²-15t, the time interval during which the velocity is negative and acceleration is positive is __________

12th Standard Maths Application of Differential Calculus English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.
2)
The tangent to the curve y² - xy + 9 = 0 is vertical when
3)
The minimum value of the function |3 - x| + 9 is
4)
The point of inflection of the curve y = (x - 1)³ is
5)
The point on the curve y = x² is the tangent parallel to X-axis is __________

12th Standard Maths Differentials and Partial Derivatives English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is
2)
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
3)
The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is
4)
If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is
5)
If y = x⁴ - 10 and if x changes from 2 to 1.99, the approximate change in y is ________

12th Standard Maths Applications of Integration English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is
2)
The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is
3)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
4)
If \(\int _{ 0 }^{ a }{ \frac { 1 }{ 4+{ x }^{ 2 } } dx=\frac { \pi }{ 8 } } \) then a is
5)
\(\text { The value of } \int_{0}^{\frac{2}{3}} \frac{d x}{\sqrt{4-9 x^{2}}} \text { is }\)

12th Standard Maths Applications of Integration English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The area between y² = 4x and its latus rectum is
2)
The value of \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+{ 5 }^{ cos\ x } } } \) is
3)
The volume of solid of revolution of the region bounded by y²= x(a − x) about x-axis is
4)
The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \) is
5)
The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x\ cos \ x \ dx } \) is

12th Standard Maths Ordinary Differential Equations English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively
2)
The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is
3)
The solution of the differential equation \(2x\frac{dy}{dx}-y=3\) represents
4)
The degree of the differential equation \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is
5)
The solution of the differential equation \(\frac { dy }{ dx } +\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } =0\) is

12th Standard Maths Ordinary Differential Equations English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is
2)
The general solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } \) is
3)
The solution of \(\frac{d y}{d x}+p(x) y=0\) is
4)
If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When
5)
If sin x is the integrating factor of the linear differential equation \(\frac { dy }{ dx } +Py=Q,\) then P is

12th Standard Maths Probability Distributions English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct
2)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
3)
If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?
4)
Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E(X) is
5)
If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).

12th Standard Maths Probability Distributions Equations English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.
2)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are
3)
Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are
4)
On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is
5)
If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is

12th Standard Maths Discrete Mathematics English Medium Free Online Test One Mark Questions 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
A binary operation on a set S is a function from
2)
The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on
3)
If a * b=\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) on the real numbers then * is
4)
Which one is the inverse of the statement (pVq)➝(pΛq)?
5)
Which one of the following is incorrect? For any two propositions p and q, we have

12th Standard Maths Discrete Mathematics Equations English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021 - by Question Bank Software - Nov 05, 2020 - View & Read

1)
Subtraction is not a binary operation in
2)
If a compound statement involves 3 simple statements, then the number of rows in the truth table is
3)
Which one is the contrapositive of the statement (pVq)⟶r?
4)
The proposition p ∧ (¬p ∨ q) is
5)
If * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is _____________

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Three - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________
2)
The domain of cos^-1(x² - 4) is______
3)
The auxiliary circle of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 16 } \) = 1 is __________
4)
The locus of the point of intersection of perpendicular tangents of the parabola y² = 4ax is
5)
The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is ___________

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Two - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I₂, then B =
2)
If \(\rho\) (A) = r then which of the following is correct?
3)
In the system of liner equations with 3 unknowns If \(\rho\) (A) = \(\rho\) ([A|B]) =1, the system has ________
4)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
5)
If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if _________

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Three - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
2)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
3)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
4)
If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA⁻¹ = A, then λ is
5)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Four - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The augmented matrix of a system of linear equations is \(\left[\begin{array}{cccc} 1 & 2 & 7 & 3 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & \lambda-7 & \mu+5 \end{array}\right]\). The system has infinitely many solutions if
2)
If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂+ 4z₁z₃+ z₂z₃| = 12, then the value of |z₁+z₂+z₃| is
3)
When the eccentricity of a ellipse becomes zero, then it becomes a __________
4)
The angle between the lines \(\frac{x-2}{3}=\frac{y+1}{-2}, z=2 \text { and } \frac{x-1}{1}=\frac{2 y+3}{3}=\frac{z+5}{2}\) is
5)
The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Four - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I₂, then B =
2)
If \(\sqrt { a+ib } \) = x + iy, then possible value of \(\sqrt { a-ib }\) is ___________
3)
The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) = 1 lies on _________
4)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
5)
The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Five - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If 0 ≤ θ ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is
2)
The solution of the equation |z| - z = 1 + 2i is
3)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
4)
The circle x²+ y²= 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if
5)
The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x²− 9y²= 144 are the roots of x²− (a + b)x − 4 = 0, then the value of (a+b) is

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Five - by Question Bank Software - Nov 05, 2020 - View & Read

1)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
2)
The principal argument of \(\cfrac { 3 }{ -1+i } \) is
3)
The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________
4)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
5)
For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Six - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
3)
If \(\alpha \) and \(\beta \) are the roots of x²+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is
4)
If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________
5)
lf the root of the equation x³ + bx²+ cx - 1 = 0 form an lncreasing G.P, then ___________

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Six - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A^TA⁻¹ is symmetric, then A² =
2)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
3)
The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is
4)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
5)
If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Seven - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If 0 ≤ θ ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is
2)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
3)
If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals
4)
The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________
5)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Seven - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
2)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
3)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
4)
If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________
5)
\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Eight - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
2)
Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is
3)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
4)
If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0
5)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Eight - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is
2)
sin (tan^-1x), |x| < 1 is equal to
3)
The value of \({ cos }^{ -1 }\left( \cos\cfrac { 5\pi }{ 3 } \right) +sin^{ -1 }\left( \sin\cfrac{5\pi }{ 3 } \right) \) is ______________
4)
The equation of the normal to the circle x²+ y²− 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is
5)
Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Nine - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is
2)
If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0
3)
The equation \(\tan ^{-1} x-\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)has
4)
In a \(\Delta ABC\) if C is a right angle, then \({ tan }^{ -1 }\left( \frac { a }{ b+c } \right) +{ tan }^{ -1 }\left( \frac { b }{ c+a } \right) =\) ________
5)
If x = r cos θ, y = r sin, then \(\frac { \partial r }{ \partial x } \) = ....................

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Nine - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A^TA⁻¹ is symmetric, then A² =
2)
If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is
3)
If a = 3 + i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and - az are ___________
4)
If z₁, z₂, z₃ are the vertices of a parallelogram, then the fourth vertex z₄ opposite to z₂ is _____
5)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

12th Standard Maths English Medium Free Online Test 1 Mark Questions 2020 - Part Ten - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
2)
If A is a matrix of order m \(\times\) n, then \(\rho\) (A) is _________
3)
If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is
4)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
5)
Tangents are drawn to the hyperbola \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

12th Standard Maths English Medium Free Online Test One Mark Questions with Answer Key 2020 - Part Ten - by Question Bank Software - Nov 05, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If z is a complex number such that \(z \in \mathbb{C} \backslash \mathbb{R}\) and \(z+\frac { 1 }{ z } \epsilon R\), then |z| is
3)
\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to
4)
If x + y = k is a normal to the parabola y² = 12x, then the value of k is
5)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is

12th Standard Maths English Medium Model 5 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2
2)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]
3)
If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc
4)
If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x²= sin 2a
5)
A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

12th Standard Maths English Medium Model 5 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
2)
The prices of three commodities A, B and C are Rs. x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B. Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn Rs. 15,000, Rs. 1,000 and Rs. 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)
3)
If ax² + bx + c is divided by x + 3, x − 5, and x − 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)
4)
Investigate for what values of λ and μ the system of linear equations x + 2y + z = 7 , x + y + λz = μ , x + 3y − 5z = 5 has
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions
5)
If the system of equations px + by + cz = 0, ax + qy + cz = 0, ax + by + rz = 0 has a non-trivial solution and p ≠ a, q ≠ b, r ≠ c, prove that \(\frac { p }{ p-a } +\frac { q }{ q-b } +\frac { r }{ r-c } =2\).

12th Standard Maths English Medium Sample 5 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.
2)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]
3)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
4)
Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\)
5)
A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

12th Standard Maths English Medium Sample 5 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.
2)
In a T20 match, a team needed just 6 runs to win with 1 ball left to go in the last over. The last ball was bowled and the batsman at the crease hit it high up. The ball traversed along a path in a vertical plane and the equation of the path is y = ax² + bx + c with respect to a xy-coordinate system in the vertical plane and the ball traversed through the points (10, 8), (20, 16) (40, 22) can you conclude that the team won the match?
Justify your answer. (All distances are measured in metres and the meeting point of the plane of the path with the farthest boundary line is (70, 0).)
3)
Test for consistency of the following system of linear equations and if possible solve:
4x − 2y + 6z = 8, x + y − 3z = −1, 15x − 3y + 9z = 21.
4)
Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.
5)
Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{matrix} \right] \)

12th Standard Maths English Medium Important 5 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.
2)
Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.
3)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
4)
Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\)
5)
The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.

12th Standard Maths English Medium Important 5 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.
2)
The prices of three commodities A, B and C are Rs. x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B. Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn Rs. 15,000, Rs. 1,000 and Rs. 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)
3)
A boy is walking along the path y = ax² + bx + c through the points (−6, 8), (−2, −12) and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)
4)
Find the value of k for which the equations
kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution
5)
Solve the following system of homogenous equations.
3x + 2y + 7z = 0, 4x − 3y − 2z = 0, 5x + 9y + 23z = 0

12th Standard Maths English Medium Model 3 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
2)
Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.
3)
Solve: (x-1)⁴+(x-5)⁴= 82
4)
Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)
5)
For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.

12th Standard Maths English Medium Model 3 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1
2)
Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.
3)
Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)
4)
Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.
5)
If |z| = 1, show that \(2\le \left| { z }^{ 2 }-3 \right| \le 4\)

12th Standard Maths English Medium Sample 3 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
2)
Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.
3)
Solve: (x-1)⁴+(x-5)⁴= 82
4)
Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)
5)
Find the value of p so that 3x + 4y - p = 0 is a tangent to the circle x² +y²- 64 = 0.

12th Standard Maths English Medium Sample 3 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).
2)
Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.
3)
Test for consistency of the following system of linear equations and if possible solve:
x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.
4)
Solve the following system of linear equations by matrix inversion method :
2x − y = 8 , 3x + 2y = −2.
5)
The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

12th Standard Maths English Medium Important 3 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
2)
Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.
3)
Solve: 2x+2^x-1+2^x-2= 7x+7^x-1+7^x-2
4)
Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)
5)
Find the equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13.

12th Standard Maths English Medium Important 3 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Sep 03, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = (adj A)A = |A|I₂.
2)
Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.
3)
Solve the following system of linear equations, by Gaussian elimination method : 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1.
4)
Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.
5)
If \(\frac { 1+z }{ 1-z } =cos2\theta +isin2\theta \), show that z = i tan\(\theta\)

12th Standard Mathematics English Medium Model 2 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Aug 29, 2020 - View & Read

1)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
2)
If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ = 0
3)
Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.
4)
Find the principal value of sin^-1(-1).
5)
Prove that \(2{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) ={ tan }^{ -1 }\left( \frac { 12 }{ 5 } \right) \)

12th Standard Mathematics English Medium Model 2 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Aug 29, 2020 - View & Read

1)
Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.
2)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
3)
Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0
4)
Evaluate the following if z = 5−2i and w = −1+3i
z + w
5)
If the area of the triangle formed by the vertices z, iz and z + iz is 50 square units, find the value of |z|

12th Standard Mathematics English Medium Sample 2 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Aug 29, 2020 - View & Read

1)
Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
2)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
3)
Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.
4)
Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)
5)
Find the length of the tangent from (2, -3) to the circle x² + y²- 8x - 9y + 12 = 0.

12th Standard Mathematics English Medium Sample 2 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Aug 29, 2020 - View & Read

1)
Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).
2)
Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right] \)
3)
Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)
4)
If z₁= 3 - 2i and z₂= 6 + 4i, find \(\frac { { z }_{ 1 } }{ z_{ 2 } } \) in the rectangular form.
5)
Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:
|z - 4| = 16

12th Standard Mathematics English Medium Important 2 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Aug 29, 2020 - View & Read

1)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
2)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
3)
Find the modules of (1+ 3i)³
4)
Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)
5)
Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)

12th Standard Mathematics English Medium Important 2 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Aug 29, 2020 - View & Read

1)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
2)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
3)
Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} -2 & 2 & -1 \\ 0 & 5 & 1 \\ 0 & 0 & 0 \end{matrix} \right] \)
4)
Find the following \(\left| \frac { 2+i }{ -1+2i } \right| \)
5)
Write in polar form of the following complex numbers
\(2+i2\sqrt { 3 } \)

12th Standard Mathematics English Medium Model 1 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Aug 25, 2020 - View & Read

1)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________
2)
If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--
3)
If \(\rho\) (A) ≠ \(\rho\) ([AIB]), then the system is _____________
4)
If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________
5)
The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) = 1 lies on _________

12th Standard Mathematics English Medium Model 1 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Aug 25, 2020 - View & Read

1)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =
2)
If 0 ≤ θ ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is
3)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
4)
The principal argument of \(\cfrac { 3 }{ -1+i } \) is
5)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is

12th Standard Mathematics English Medium Sample 1 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Aug 25, 2020 - View & Read

1)
If \(\rho\) (A) = r then which of the following is correct?
2)
In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______
3)
If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________
4)
The modular of \(\frac { (-1+i)(1-i) }{ 1+i\sqrt { 3 } } \) is ______
5)
If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

12th Standard Mathematics English Medium Sample 1 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Aug 25, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
2)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =
3)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
4)
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I
5)
If z is a complex number such that \(z \in \mathbb{C} \backslash \mathbb{R}\) and \(z+\frac { 1 }{ z } \epsilon R\), then |z| is

12th Standard Mathematics English Medium Important 1 Mark Creative Questions (New Syllabus) 2020 - by Question Bank Software - Aug 25, 2020 - View & Read

1)
Which of the following is not an elementary transformation?
2)
If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________
3)
If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if _________
4)
If x = cos θ + i sin θ, then xⁿ+ \(\frac { 1 }{ { x }^{ n } } \) is ______
5)
If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________

12th Standard Mathematics English Medium Important 1 Mark Book Back Questions (New Syllabus) 2020 - by Question Bank Software - Aug 25, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If \(\rho\) (A) = \(\rho\)([A| B]), then the system AX = B of linear equations is
3)
If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is
4)
If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals
5)
A polynomial equation in x of degree n always has

12th Standard Mathematics English Medium All Chapter One Marks Book Back and Creative Questions 2020 - by Question Bank Software - Mar 07, 2020 - View & Read

1)
If 0 ≤ θ ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is
2)
If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is
3)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
4)
In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z is non zero then the system is ______
5)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

12th Standard Mathematics English Medium All Chapter Two Marks Book Back and Creative Questions 2020 - by Question Bank Software - Mar 07, 2020 - View & Read

1)
Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)
2)
Reduce the matrix \(\left[ \begin{matrix} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{matrix} \right] \) to a row-echelon form.
3)
Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.
4)
Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).
5)
Find z⁻¹, if z = (2 + 3i) (1− i).

12th Standard Mathematics English Medium All Chapter Three Marks Book Back and Creative Questions 2020 - by Question Bank Software - Mar 07, 2020 - View & Read

1)
Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.
2)
Solve the following system of linear equations by matrix inversion method:
2x + 5y = −2, x + 2y = −3
3)
Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.
4)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
5)
If z₁= 3, z₂= -7i, and z₃= 5 + 4i, show that z₁(z₂+ z₃) = z₁z₂+ z₁z₃

12th Standard Mathematics English Medium All Chapter Five Marks Book Back and Creative Questions 2020 - by Question Bank Software - Mar 07, 2020 - View & Read

1)
Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.
2)
Solve the following system of linear equations by matrix inversion method:
2x + 3y − z = 9, x + y + z = 9, 3x − y − z = −1
3)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.
4)
Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.
5)
Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r

12th Standard Mathematics Public Exam Model Question Paper III 2019 - 2020 - by Question Bank Software - Feb 21, 2020 - View & Read

1)
If 0 ≤ θ ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is
2)
If A = [2 0 1] then the rank of AA^T is ______
3)
If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________
4)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
5)
If ax² + bx + c = 0, a, b, c \(\in\) R has no real zeros, and if a + b + c < 0, then __________

12th Standard Mathematics Public Exam Model Question Paper II 2019 - 2020 - by Question Bank Software - Feb 21, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
2)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
3)
\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________
4)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
5)
If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

12th Maths - Discrete Mathematics - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
a*b = a + 3ab − 5b²; ∀a,b∈Z
2)
Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)
3)
How many rows are needed for following statement formulae?
\(p \vee \neg t \wedge(p \vee \neg s)\)
4)
Determine whether ∗ is a binary operation on the sets given below.
(a*b) = a√b is binary on R
5)
Let A = {a +\(\sqrt5\) b : a,b∈Z}. Check whether the usual multiplication is a binary operation on A.

12th Maths - Probability Distributions - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.
2)
An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.
3)
A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)
4)
The cumulative distribution function of a discrete random variable is given by

Find
(i) the probability mass function
(ii) P(X < 1 ) and
(iii) P(X \(\geq\)2)
5)
A random variable X has the following probability mass function.

x 1 2 3 4 5

f(x) k² 2k² 3k² 2k 3k

Find
(i) the value of k
(ii) P(2 \(\le\) X < 5)
(iii) P(3 < X )

12th Maths - Ordinary Differential Equations - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { dy }{ dx } +xy=cotx\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)
3)
For each of the following differential equations, determine its order, degree (if exists)
\({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)
4)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)
5)
Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.

12th Maths - Applications of Integration - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.
2)
Evaluate the following:
\(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx } \)
3)
Evaluate \(\int ^\frac {\pi}{2}_{0} \)( sin² x + cos⁴ x ) dx
4)
Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)
5)
Evaluate the following
\(\int _{ 0 }^{ 1 }{ { x }^{ 2 }{ (1-x) }^{ 3 }dx } \)

12th Maths - Differentials and Partial Derivatives - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Use the linear approximation to find approximate values of \({ (123) }^{ \frac { 2 }{ 3 } }\)
2)
Find a linear approximation for the following functions at the indicated points.
\(h(x)=\frac{x}{x+1}, x_{0}=1\)
3)
Find ∆f and df for the function f for the indicated values of x, ∆x and compare
4)
Let g(x, y) = \(\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } } \) for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) g(x, y) = 0 along every line y = mx, m ∈ R

12th Maths - Application of Differential Calculus - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) = 100 × (1− 0.1t)², 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?
2)
A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
3)
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
4)
Compute the value of 'c' satisfied by the Rolle’s theorem for the function f (x) = x² (1 - x)², x ∈ [0,1]
5)
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
\(f(x)=|\frac{1}{x}|, x\in [-1,1]\)

12th Maths - Applications of Vector Algebra - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \), \(\hat { 2i } -\hat { j } +\hat { 2k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)
2)
Determine whether the three vectors \(2\hat { i } +3\hat { j } +\hat { k } \), \(\hat { i } -2\hat { j } +2\hat { k } \) and \(\hat { 3i } +\hat { j } +3\hat { k } \) are coplanar.
3)
Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector \(4\hat { i } +3\hat { j } -7\hat { k } \) and parallel to the vector \(2\hat { i } -6\hat { j } +7\hat { k } \).
4)
Find the angle between the line \(\vec { r } =(2\hat { i } -\hat { j } +\hat { k } )+t(\hat { i } +2\hat { j } -2\hat { k } )\) and the plane \(\vec { r } =(6\hat { i } +3\hat { j } +2\hat { k } )=8\)
5)
Find the acute angle between the following lines
2x = 3y = −z and 6x = − y = −4z.

12th Maths - Two Dimensional Analytical Geometry II - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Examine the position of the point (2, 3) with respect to the circle x²+ y²− 6x − 8y + 12 = 0.
2)
Find the equation of the circle with centre (2, -1) and passing through the point (3, 6) in standard form.
3)
Identify the type of the conic for the following equations :
11x²−25y²−44x+50y−256 = 0
4)
Find centre and radius of the following circles.
x²+ y²+ 6x − 4y + 4 = 0
5)
Find centre and radius of the following circles.
2x²+2y²−6x+4y+2 = 0

12th Maths - Inverse Trigonometric Functions - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Find the principal value of sin^-1(2), if it exists.
2)
Find the period and amplitude of y = sin 7x
3)
Find the value of
\(2{ cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) +{ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) \)
4)
Find the value of sec⁻¹\(\left( -\frac { 2\sqrt { 3 } }{ 3 } \right) \)
5)
Prove that \(\frac{\pi}{2}\le sin^{-1}x+2 cos^{-1} x\le\frac{3\pi}{2}\)

12th Maths - Theory of Equations - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Construct a cubic equation with roots 1, 2 and 3
2)
If α, β, γ and \(\delta\) are the roots of the polynomial equation 2x⁴+ 5x³− 7x²+ 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).
3)
Show that the equation 2x²− 6x +7 = 0 cannot be satisfied by any real values of x.
4)
Show that if p, q, r are rational the roots of the equation x²− 2px + p²− q²+ 2qr − r²= 0 are rational.
5)
Obtain the condition that the roots of x³+ px²+ qx + r = 0 are in A.P.

12th Maths - Complex Numbers - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form
2)
If z₁= 3 - 2i and z₂= 6 + 4i, find \(\frac { { z }_{ 1 } }{ z_{ 2 } } \) in the rectangular form.
3)
Find the modulus of the following complex numbers
\(\frac { 2i }{ 3+4i } \)
4)
Find the square roots of 4+3i
5)
Show that the following equations represent a circle, and find its centre and radius \(\left| z-2-i \right| =3\)

12th Maths - Application of Matrices and Determinants - Two Marks Study Materials - by 8682895000 - Jan 22, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
3)
If A is symmetric, prove that then adj A is also symmetric.
4)
If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A⁻¹.
5)
Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 3 & 2 & 5 \\ 1 & 1 & 2 \\ 3 & 3 & 6 \end{matrix} \right] \)

12th Maths - Full Portion Five Marks Question Paper - by 8682895000 - Jan 20, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
2)
In a T20 match, a team needed just 6 runs to win with 1 ball left to go in the last over. The last ball was bowled and the batsman at the crease hit it high up. The ball traversed along a path in a vertical plane and the equation of the path is y = ax² + bx + c with respect to a xy-coordinate system in the vertical plane and the ball traversed through the points (10, 8), (20, 16) (40, 22) can you conclude that the team won the match?
Justify your answer. (All distances are measured in metres and the meeting point of the plane of the path with the farthest boundary line is (70, 0).)
3)
Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.
4)
Solve the following systems of linear equations by Cramer’s rule:
3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.
5)
For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.

12th Maths - Full Portion Three Marks Questions - by 8682895000 - Jan 20, 2020 - View & Read

1)
If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.
2)
Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method.
3)
Find the rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).
4)
The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.
5)
If (x₁+ iy₁)(x₂+ iy₂)(x3 + iy₃)...(x_n+ iy_n) = a + ib, show that
(x₁²+ y₁²)(x₂²+ y₂²)(x₃²+ y₃²)...(x_n²+ y_n²) = a²+ b²

12th Maths - Full Portion Two Marks Question Paper - by 8682895000 - Jan 20, 2020 - View & Read

1)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
2)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
3)
Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0
4)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
5)
Find z⁻¹, if z = (2 + 3i) (1− i).

REVISION TEST - by Win Academy - Jan 18, 2020 - View & Read

1)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
2)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
3)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
4)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
5)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is

12th Maths -Public Exam Model Question Paper 2019 - 2020 - by Question Bank Software - Jan 17, 2020 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If A = [2 0 1] then the rank of AA^T is ______
3)
If x = cos θ + i sin θ, then xⁿ+ \(\frac { 1 }{ { x }^{ n } } \) is ______
4)
A polynomial equation in x of degree n always has
5)
If the equation ax²+ bx+c = 0(a > 0) has two roots ∝ and β such that ∝ <- 2 and β > 2, then __________

12th Maths - Applications of Integration Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)
2)
The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is
3)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
4)
The value of \(\int _{ \frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \sqrt { \frac { 1-cos2x }{ 2x } } } \) dx is __________
5)
The area enclosed by the curve y = \(\frac { { x }^{ 2 } }{ 2 } \) , the x - axis and the lines x = 1, x = 3 is __________

12th Maths - Differentials and Partial Derivatives Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
2)
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
3)
If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________
4)
If f(x, y, z) = sin (xy) + sin (yz) + sin (zx) then f_xx is _____________
5)
If f(x, y) = 2x²- 3xy + 5y + 7 then f(0, 0) and f(1, 1) is _____________

12th Standard Maths - Applications of Vector Algebra Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
2)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to
3)
The coordinates of the point where the line \(\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(-\hat { i } +4\hat { j } )\) meets the plane \(\vec { r } .(\hat { i } +\hat { j } -\hat { k } )\) = 3 are
4)
The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________
5)
If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________

Plus 2 Maths - Two Dimensional Analytical Geometry-II Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
2)
The equation of the normal to the circle x²+ y²− 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is
3)
If x + y = k is a normal to the parabola y² = 12x, then the value of k is
4)
y²- 2x - 2y + 5 = 0 is a _________
5)
If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________

12th Stateboard Maths - Probability Distributions Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct
2)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
3)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are
4)
If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?
5)
If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).

12th Maths Chapter 12 Discrete Mathematics Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
Subtraction is not a binary operation in
2)
Which one of the following statements has truth value F?
3)
Which one is the contrapositive of the statement (pVq)⟶r?
4)
The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________
5)
Which of the following is a contradiction?

12th Maths - Ordinary Differential Equations Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is
2)
The solution of the differential equation \(2x\frac{dy}{dx}-y=3\) represents
3)
If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When
4)
The solution of sec²x tan y dx + sec²y tan x dy = 0 is _________
5)
The solution of (x²- ay)dx = (ax - y²)dy is ___________

12th Maths - Application of Differential Calculus Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by
2)
The tangent to the curve y² - xy + 9 = 0 is vertical when
3)
The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is
4)
The point on the curve y = x² is the tangent parallel to X-axis is __________
5)
The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________

12th Maths - Inverse Trigonometric Functions Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for
2)
If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is
3)
The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________
4)
\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________
5)
If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________

12th Maths - Theory of Equations Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
2)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is
3)
If x is real and \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then ________
4)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
5)
If ∝, β, ૪ are the roots of the equation x³-3x+11 = 0, then ∝+β+૪ is __________.

12th Maths - Complex Numbers Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
2)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
3)
If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂+ 4z₁z₃+ z₂z₃| = 12, then the value of |z₁+z₂+z₃| is
4)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______
5)
If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________

12th Maths - Application of Matrices and Determinants Model Question Paper 1 - by 8682895000 - Jan 06, 2020 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
3)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
4)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
5)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

12th Maths - Discrete Mathematics Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
Which one of the following statements has the truth value T?
2)
Which one of the following is not true?
3)
The number of binary operations that can be defined on a set of 3 elements is _____________
4)
The identity element in the group {R - {1},x} where a * b = a + b - ab is __________
5)
If p is true and q is false, then which of the following is not true?
(1) p ⟶ q is F
(2) p v q is T
(3) p ∧ q is F
(4) p ⇔ q is F

12th Maths - Probability Distributions Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are
2)
If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).
3)
A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)
4)
A random variable X has the following probability mass function.

x 1 2 3 4 5

f(x) k² 2k² 3k² 2k 3k

Find
(i) the value of k
(ii) P(2 \(\le\) X < 5)
(iii) P(3 < X )
5)
Find the binomial distribution function for each of the following.
(i) Five fair coins are tossed once and X denotes the number of heads.
(ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

12th Maths - Ordinary Differential Equations Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
The general solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } \) is
2)
If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When
3)
The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1, 1). Then the equation of the curve is
4)
The I.F. of cosec x \(\frac{dy}{dx}+y\) sec²x = 0 is ___________
5)
The differential equation associated with the family of concentric circles having their centres at the origin is _________.

12th Maths - Applications of Integration Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)
2)
If \(f(x)=\int_{1}^{x} \frac{e^{\sin u}}{u} d u, x>1 \text { and }\int_{1}^{3} \frac{e^{\sin x^{2}}}{x} d x=\frac{1}{2}[f(a)-f(1)]\), then one of the possible value of a is
3)
The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x\ cos \ x \ dx } \) is
4)
The area enclosed by the curve y = \(\frac { { x }^{ 2 } }{ 2 } \) , the x - axis and the lines x = 1, x = 3 is __________
5)
The area enclosed by the curve y² = 4x, the x-axis and its latus rectum is ________ sq.units.

12th Maths - Differentials and Partial Derivatives Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to
2)
If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to
3)
If f(x, y, z) = sin (xy) + sin (yz) + sin (zx) then f_xx is _____________
4)
If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............
5)
If u = log \(\left( \frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \right) \) then
(1) u is a homogeneous function
(2) \(x\frac { { \partial }u }{ \partial { x } } +y\frac { { \partial }u }{ { \partial y } } \) = 0
(3) \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \) is a homogeneous function
(4) \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \) is a homogeneous function of degree 0.

12th Maths - Application of Differential Calculus Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is
2)
The function sin⁴ x + cos⁴x is increasing in the interval
3)
One of the closest points on the curve x² - y² = 4 to the point (6, 0) is
4)
The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________
5)
The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

Applications of Vector Algebra Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
If \([\vec{a}, \vec{b}, \vec{c}]=1\), then the value of \(\frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{c} \times \vec{a})}{(\vec{a} \times \vec{b}) \cdot \vec{c}}+\frac{\vec{c} \cdot(\vec{a} \times \vec{b})}{(\vec{c} \times \vec{b}) \cdot \vec{a}}\) is
2)
If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are
3)
The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________
4)
The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is ___________
5)
The distance from the origin to the plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 2i } -\overset { \wedge }{ j } +5\overset { \wedge }{ k } \right) =7\) is ______________

12th Maths - Two Dimensional Analytical Geometry-II Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is
2)
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)
3)
Equation of tangent at (-4, -4) on x² = -4y is _____________
4)
y²- 2x - 2y + 5 = 0 is a _________
5)
The length of major and minor axes of 4x² + 3y² = 12 are ____________

12th Maths - Inverse Trigonometric Functions Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
If \(\cot ^{-1}(\sqrt{\sin \alpha})+\tan ^{-1}(\sqrt{\sin \alpha})=u\), then cos2u is equal to
2)
If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to
3)
If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is
4)
sin (tan^-1x), |x| < 1 is equal to
5)
The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

12th Maths - Theory of Equations Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
2)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
3)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
4)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
5)
If the equation ax²+ bx+c = 0(a > 0) has two roots ∝ and β such that ∝ <- 2 and β > 2, then __________

12th Maths - Complex Numbers Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
3)
If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is
4)
If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is
5)
The value of (1+i) (1+i²) (1+i³) (1+i⁴) is ____________

12th Maths - Application of Matrices and Determinants Model Question Paper - by Question Bank Software - Jan 02, 2020 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
4)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
5)
The rank of any 3 \(\times\) 4 matrix is
(1) May be 1
(2) May be 2
(3) May be 3
(4) Maybe 4

12th Maths - Discrete Mathematics Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
A binary operation on a set S is a function from
2)
In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?
3)
Which one is the contrapositive of the statement (pVq)⟶r?
4)
Which of the following is a tautology?
5)
The identity element in the group {R - {1},x} where a * b = a + b - ab is __________

12th Maths - Probability Distributions Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is
2)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
3)
Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are
4)
If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).
5)
Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is

12th Maths - Ordinary Differential Equations Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is
2)
The solution of the differential equation \(2x\frac{dy}{dx}-y=3\) represents
3)
The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is
4)
The solution of (x²- ay)dx = (ax - y²)dy is ___________
5)
The transformation y = vx reduces \(\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x } \) __________

12th Maths - Applications of Integration Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is
2)
The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is
3)
\(\text { The value of } \int_{0}^{\frac{2}{3}} \frac{d x}{\sqrt{4-9 x^{2}}} \text { is }\)
4)
If \(\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) } \) then __________
5)
The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is __________

12th Maths - Differentials and Partial Derivatives Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to
2)
If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to
3)
The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is
4)
If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________
5)
If u = x^y + y^x then u_x + u_y at x = y = 1 is _____________

12th Maths - Application of Differential Calculus Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is
2)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
3)
The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is
4)
The point on the curve y = x² is the tangent parallel to X-axis is __________
5)
The equation of the tangent to the curve y = x²-4x+2 at (4, 2) is __________

12th Maths - Applications of Vector Algebra Model Question Paper - by Question Bank Software - Dec 03, 2019 - View & Read

1)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
2)
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
3)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is
4)
The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________
5)
If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________

12th Maths - Two Dimensional Analytical Geometry II Model Question Paper - by Question Bank Software - Nov 27, 2019 - View & Read

1)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
2)
The centre of the circle inscribed in a square formed by the lines x²− 8x − 12 = 0 and y²− 14y + 45 = 0 is
3)
If P(x, y) be any point on 16x²+ 25y²= 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁ + PF₂ is
4)
If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)²+ (y + 2)²= r² , then the value of r² is
5)
If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________

12th Maths - Inverse Trigonometric Functions Model Question Paper - by Question Bank Software - Nov 27, 2019 - View & Read

1)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to
2)
If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is
3)
If the function f(x) = sin^-1(x²- 3), then x belongs to
4)
The value of \({ cos }^{ -1 }\left( \cos\cfrac { 5\pi }{ 3 } \right) +sin^{ -1 }\left( \sin\cfrac{5\pi }{ 3 } \right) \) is ______________
5)
If \({ tan }^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \frac { x-1 }{ x } \right) ={ tan }^{ -1 }\left( -7 \right) \) then x is ___________

12th Maths - Theory of Equations Model Question Paper - by Question Bank Software - Nov 27, 2019 - View & Read

1)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
2)
A polynomial equation in x of degree n always has
3)
Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________
4)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
5)
If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

12th Maths - Complex Numbers Important Questions - by Question Bank Software - Nov 21, 2019 - View & Read

1)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
2)
If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is
3)
The principal argument of the complex number \(\frac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is
4)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______
5)
If x + iy = \(\frac { 3+5i }{ 7-6i } \), they y = ___________

12th Maths - Application of Matrices and Determinants Important Questions - by Question Bank Software - Nov 21, 2019 - View & Read

1)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
2)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
3)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
4)
The augmented matrix of a system of linear equations is \(\left[\begin{array}{cccc} 1 & 2 & 7 & 3 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & \lambda-7 & \mu+5 \end{array}\right]\). The system has infinitely many solutions if
5)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________

12th Maths Half Yearly Model Question Paper 2019 - by Question Bank Software - Nov 12, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
3)
The principal argument of \(\cfrac { 3 }{ -1+i } \) is
4)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______
5)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?

12th Maths - Applications of Vector Algebra One Mark Questions with Answer - by Question Bank Software - Nov 05, 2019 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
3)
If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is
4)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to
5)
If \([\vec{a}, \vec{b}, \vec{c}]=1\), then the value of \(\frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{c} \times \vec{a})}{(\vec{a} \times \vec{b}) \cdot \vec{c}}+\frac{\vec{c} \cdot(\vec{a} \times \vec{b})}{(\vec{c} \times \vec{b}) \cdot \vec{a}}\) is

12th Maths - Two Dimensional Analytical Geometry II One Mark Questions with Answer - by Question Bank Software - Nov 05, 2019 - View & Read

1)
The equation of the directrix of the parabola y²+ 4y + 4x + 2 = 0 is ___________
2)
Equation of tangent at (-4, -4) on x² = -4y is _____________
3)
If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________
4)
The eccentricity of the ellipse 9x²+ 5y² - 30y = 0 is __________
5)
The length of the latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is __________

12th Maths - Inverse Trigonometric Functions One Mark Questions with Answer - by Question Bank Software - Nov 05, 2019 - View & Read

1)
If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________
2)
If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________
3)
The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________
4)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________
5)
The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi is ___________

12th Maths - Theory of Equations One Mark Questions with Answer - by Question Bank Software - Nov 05, 2019 - View & Read

1)
A zero of x³ + 64 is
2)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
3)
A polynomial equation in x of degree n always has
4)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
5)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?

12th Maths - Complex Numbers One Mark Questions with Answer - by Question Bank Software - Nov 05, 2019 - View & Read

1)
The value of (1+i) (1+i²) (1+i³) (1+i⁴) is ____________
2)
If \(\sqrt { a+ib } \) = x + iy, then possible value of \(\sqrt { a-ib }\) is ___________
3)
If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________
4)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______
5)
If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________

12th Maths - Application of Matrices and Determinants One Mark Questions with Answer - by Question Bank Software - Nov 05, 2019 - View & Read

1)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
2)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
3)
Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________
4)
If A^T is the transpose of a square matrix A, then ___________
5)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________

12th Maths - Discrete Mathematics Five Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining'.
2)
How many rows are needed for following statement formulae?
\(p \vee \neg t \wedge(p \vee \neg s)\)
3)
How many rows are needed for following statement formulae?
(( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))
4)
Construct the truth table for \((p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)\)
5)
Verify
(i) closure property
(ii) commutative property
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the operation +₅ on Z₅ using table corresponding to addition modulo 5.

12th Maths - Probability Distributions Five Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.
2)
A pair of fair dice is rolled once. Find the probability mass function to get the number of fours.
3)
If the probability mass function f(x) of a random variable X is

x 1 2 3 4

f (x) \(\cfrac { 1 }{ 12 } \) \(\cfrac { 5 }{ 12 } \) \(\cfrac { 5 }{ 12 } \) \(\cfrac { 1 }{ 12 } \)

find (i) its cumulative distribution function, hence find
(ii) P(X ≤ 3) and,
(iii) P(X ≥ 2)
4)
A six sided die is marked ‘1’ on one face, ‘2’ on two of its faces, and ‘3’ on remaining three faces. The die is rolled twice. If X denotes the total score in two throws.
(i) Find the probability mass function.
(ii) Find the cumulative distribution function.
(iii) Find P(3 ≤ X< 6)
(iv) Find P(X ≥ 4) .
5)
Find the probability mass function f(x) of the discrete random variable X whose cumulative distribution function F(x) is given by

Also find
(i) P(X < 0) and
(ii) P(\(X \geq-1)\)

12th Maths - Ordinary Differential Equations Five Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Form the differential equation by eliminating the arbitrary constants A and B from y = A cos x + B sin x.
2)
Find the differential equation of the family of circles passing through the points (a, 0) and (−a, 0).
3)
Find the differential equation of the family of all ellipses having foci on the x -axis and centre at the origin.
4)
Find the particular solution of (1+ x³)dy − x² ydx = 0 satisfying the condition y(1) = 2.
5)
Solve y' = sin² (x − y + 1 ).

12th Maths - Applications of Integration Five Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Evaluate \(\int _{ 0 }^{ 1 }{ x^3dx } \), as the limit of a sum.
2)
Evaluate\(\int _{ 1 }^{ 4 }{ ({ 2x }^{ 2 }+3) } \) dx, as the limit of a sum
3)
Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.
4)
Evaluate: \(\\ \\ \int _{ -1 }^{ 1 }{ { e }^{ -\lambda x }(1-{ x }^{ 2 }) } dx\)
5)
Evaluate \(\int ^\frac {\pi}{2}_{0} \)( sin² x + cos⁴ x ) dx

12th Maths - Application of Differential Calculus Five Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Find the points of x the curve y = x³ − 3x² + x − 2 at which the tangent is parallel to the line y = x
2)
Find the equation of the tangent and normal to the Lissajous curve given by x = 2cos 3t and y = 3sin 2t, t ∈ R
3)
Expand log(1+ x) as a Maclaurin’s series upto 4 non-zero terms for –1 < x ≤ 1.
4)
Expand tan x in ascending powers of x upto 5^th power for \(-\frac{\pi}{2} <x<\frac{\pi}{2}\)
5)
Find the intervals of monotonicity and hence find the local extrema for the function f(x) = x² − 4x + 4

12th Maths - Differentials and Partial Derivatives Five Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Let f, g : (a, b)→R be differentiable functions. Show that d(fg) = fdg + gdf
2)
Let g(x) = x² + sin x. Calculate the differential dg.
3)
If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?
4)
Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)
5)
Let F(x, y) = x³ y + y²x + 7 for all (x, y)∈ R². Calculate \(\frac { \partial F }{ \partial x } \)(-1, 3) and \(\frac { \partial F }{ \partial y } \)(-2, 1).

12th Maths - Discrete Mathematics Three Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.
2)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_e = the set of all even integers
3)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_o = the set of all odd integers
4)
Verify
(i) closure property
(ii) commutative property, and
(iii) associative property of the following operation on the given set. (a*b) = a^b;∀a, b∈N (exponentiation property)
5)
Determine whether ∗ is a binary operation on the sets given below.
a*b = b = a.|b| on R

12th Maths - Probability Distributions Three Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Suppose two coins are tossed once. If X denotes the number of tails,
(i) write down the sample space
(ii) find the inverse image of 1
(iii) the values of the random variable and number of elements in its inverse images
2)
Suppose a pair of unbiased dice is rolled once. If X denotes the total score of two dice, write down
(i) the sample space
(ii) the values taken by the random variable X,
(iii) the inverse image of 10, and
(iv) the number of elements in inverse image of X.
3)
An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen, find the values taken by the random variable X and its number of inverse images
4)
Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.
5)
Find the constant C such that the function \(f(x)= \begin{cases}C x^{2} & 1 is a density function, and compute
(i) P(1.5 < X < 3.5)
(ii) P(X ≤ 2)
(iii) P(3 < X )

12th Maths - Ordinary Differential Equations Three Marks Questions Paper - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Determine the order and degree (if exists) of the following differential equations:
\(\frac { dy }{ dx } =x+y+5\)
2)
Determine the order and degree (if exists) of the following differential equations:
\({ \left( \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } \right) }^{ 3 }+4{ \left( \frac { dy }{ dx } \right) }^{ 7 }+6y=5cos3x\)
3)
Determine the order and degree (if exists) of the following differential equations:
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)
4)
Determine the order and degree (if exists) of the following differential equations:
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)
5)
Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0

12th Maths - Applications of Integration Three Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\) using the Riemann sums corresponding to 5 subintervals of equal width and applying
(i) left-end rule
(ii) right-end rule
(iii) the mid-point rule.
2)
Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)
3)
Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)
4)
Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)
5)
Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log_e2.

12th Maths - Differentials and Partial Derivatives Three Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
Find the linear approximation for f(x) = \(\sqrt { 1+x } ,x\ge -1\) at x₀ = 3. Use the linear approximation to estimate f(3.2)
2)
Use linear approximation to find an approximate value of \(\sqrt { 9.2 } \) without using a calculator.
3)
Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.
4)
A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.
5)
Let f (x,y) = \(\frac { 3x-5y+8 }{ { x }^{ 2 }+{ y }^{ 2 }+1 } \) for all (x, y) ∈ R²Show that f is continuous on R²

12th Maths - Application of Differential Calculus Three Marks Questions - by Question Bank Software - Nov 04, 2019 - View & Read

1)
For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2
2)
The temperature T in celsius in a long rod of length 10 m, insulated at both ends, is a function of length x given by T = x(10 − x). Prove that the rate of change of temperature at the midpoint of the rod is zero.
3)
A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.
4)
The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.
5)
Compute the value of 'c' satisfied by the Rolle’s theorem for the function f (x) = x² (1 - x)², x ∈ [0,1]

12th Maths - Discrete Mathematics Two Marks Questions - by Question Bank Software - Nov 02, 2019 - View & Read

1)
Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
a*b = a + 3ab − 5b²; ∀a,b∈Z
2)
Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)
3)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z.
4)
Let A =\(\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)be any two boolean matrices of the same type. Find AvB and A\(\wedge\)B.
5)
Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q

12th Maths - Probability Distributions Two Marks Questions - by Question Bank Software - Nov 02, 2019 - View & Read

1)
Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.
2)
Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.
3)
A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)
4)
Find the probability mass function and cumulative distribution function of number of girl child in families with 4 children, assuming equal probabilities for boys and girls.
5)
The probability density function of X is given by \(f(x)=\begin{cases} \begin{matrix} kxe^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & for\quad x\le 0 \end{matrix} \end{cases}\) Find the value of k.

12th Maths - Ordinary Differential Equations Two Marks Questions - by Question Bank Software - Nov 02, 2019 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +\int { ydx } ={ x }^{ 3 }\)
3)
Find the differential equation of the family of all nonhorizontal lines in a plane.
4)
Form the differential equation of all straight lines touching the circle x² + y² = r².
5)
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.

12th Maths - Applications of Integration Two Marks Questions - by Question Bank Software - Nov 02, 2019 - View & Read

1)
Evaluate the following integrals as the limits of sums.
\(\int _{ 0 }^{ 1 }{ (5x+4)dx } \)
2)
Evaluate the following integrals as the limits of sums.
\(\int _{ 1 }^{ 2 }{( 4x^2-1)dx } \)
3)
Evaluate the following definite integrals:
\(\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } } \)
4)
Evaluate the following definite integrals:
\(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } } \)
5)
Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 2\pi }{ xlog\left( \frac { 3+cos\ x }{ 3-cos\ x } \right) } dx\)

12th Maths - Differentials and Partial Derivatives Two Marks Questions - by Question Bank Software - Nov 02, 2019 - View & Read

1)
Find a linear approximation for the following functions at the indicated points.
g(x) = \(g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4\)
2)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error
3)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. Find the following in calculating the area of the circular plate:
Relative error
4)
Find differential dy for each of the following function \(y=\frac { { \left( 1-2x \right) }^{ 3 } }{ 3-4x } \)
5)
Find differential dy for each of the following function
y = (3 + sin(2x)) ^2/3

12th Maths - Application of Differential Calculus Two Marks Questions - by Question Bank Software - Nov 02, 2019 - View & Read

1)
A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.
2)
A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.
3)
A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?
4)
Find the slope of the tangent to the following curves at the respective given points
y = x⁴ + 2x²− x at x = 1
5)
Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x +12y = 12.

12th Maths - Discrete Mathematics One Mark Questions with Answer - by Question Bank Software - Nov 01, 2019 - View & Read

1)
A binary operation on a set S is a function from
2)
Subtraction is not a binary operation in
3)
Which one of the following is a binary operation on N?
4)
In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?
5)
The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on

12th Maths - Probability Distributions One Mark Questions with Answer - by Question Bank Software - Nov 01, 2019 - View & Read

1)
Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct
2)
A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.
3)
Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is
4)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
5)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

12th Maths - Ordinary Differential Equations One Mark Questions with Answer - by Question Bank Software - Nov 01, 2019 - View & Read

1)
The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively
2)
The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is
3)
The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is
4)
The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is
5)
The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is

12th Maths - Applications of Integration One Mark Questions with Answer - by Question Bank Software - Nov 01, 2019 - View & Read

1)
The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is
2)
The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is
3)
If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)
4)
The area between y² = 4x and its latus rectum is
5)
The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is

12th Maths - Differentials and Partial Derivatives One Mark Questions with Answer - by Question Bank Software - Nov 01, 2019 - View & Read

1)
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is
2)
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
3)
If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to
4)
If v (x, y) = log (e^x+ e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to
5)
If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

12th Maths - Application of Differential Calculus One Mark Questions with Answer - by Question Bank Software - Nov 01, 2019 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.
3)
The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is
4)
A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by
5)
The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is

12th Standard Maths - Discrete Mathematics Model Question Paper - by Question Bank Software - Oct 30, 2019 - View & Read

1)
A binary operation on a set S is a function from
2)
Which one of the following statements has the truth value T?
3)
Which one of the following statements has truth value F?
4)
Which one is the contrapositive of the statement (pVq)⟶r?
5)
Which one of the following is incorrect? For any two propositions p and q, we have

12th Standard Maths - Probability Distributions Model Question Paper - by Question Bank Software - Oct 30, 2019 - View & Read

1)
Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct
2)
Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is
3)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
4)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
5)
Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are

12th Standard Maths - Ordinary Differential Equations Model Question Paper - by Question Bank Software - Oct 30, 2019 - View & Read

1)
The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is
2)
The general solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } \) is
3)
The solution of \(\frac{d y}{d x}+p(x) y=0\) is
4)
The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is
5)
If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When

12th Standard Maths - Applications of Integration Model Question Paper - by Question Bank Software - Oct 30, 2019 - View & Read

1)
The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is
2)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)
3)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
4)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is
5)
The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x\ cos \ x \ dx } \) is

12th Standard Maths - Differentials and Partial Derivatives Model Question Paper - by Question Bank Software - Oct 30, 2019 - View & Read

1)
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is
2)
If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to
3)
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
4)
If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to
5)
If \(f(x)=\frac{x}{x+1}\), then its differential is given by

12th Standard Maths - Application of Differential Calculus Model Question Paper - by Question Bank Software - Oct 30, 2019 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is
3)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
4)
The tangent to the curve y² - xy + 9 = 0 is vertical when
5)
What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)

12th Maths - Term II Model Question Paper - by Meera - Namakkal - Oct 29, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
3)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
4)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
5)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

12th Standard Maths - Applications of Vector Algebra Model Question Paper - by Meera - Namakkal - Oct 09, 2019 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is
3)
If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } +\hat { j } \), \(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \), then the value of \(\lambda +\mu \) is
4)
Consider the vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d} \) such that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\) = \(\vec { 0 } \) Let \({ P }_{ 1 }\) and \({ P }_{ 2 }\) be the planes determined by the pairs of vectors \(\vec { a } ,\vec { b } \) and \(\vec { c } ,\vec { d } \) respectively. Then the angle between \({ P }_{ 1 }\) and \({ P }_{ 2 }\) is
5)
If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =\hat { i } +2\hat { j } -5\hat { k } ,\vec { c } =3\hat { i } +5\hat { j } -\hat { k } ,\) then a vector perpendicular to \(\vec { a } \) and lies in the plane containing \(\vec { b } \) and \(\vec { c } \) is

12th Standard Maths - Two Dimensional Analytical Geometry-II Model Question Paper - by Meera - Namakkal - Oct 09, 2019 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The radius of the circle 3x²+ by²+ 4bx − 6by + b² = 0 is
3)
The equation of the normal to the circle x²+ y²− 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is
4)
The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is
5)
If x + y = k is a normal to the parabola y² = 12x, then the value of k is

12th Standard Maths - Inverse Trigonometric Functions Model Question Paper - by Meera - Namakkal - Oct 01, 2019 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for
3)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
4)
\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation
5)
sin (tan^-1x), |x| < 1 is equal to

12th Standard Maths - Theory of Equations Model Question Paper - by Meera - Namakkal - Oct 01, 2019 - View & Read

1)
A zero of x³ + 64 is
2)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
3)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is
4)
For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________
5)
If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

12th Maths - Applications of Vector Algebra Three Marks Questions - by Question Bank Software - Sep 30, 2019 - View & Read

1)
Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \), \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.
2)
Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)
3)
Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + z + 1 = 0 and perpendicular to the plane x + 2y - 3z + 6 = 0
4)
Find the angle between the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0
5)
If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)

12th Maths - Two Dimensional Analytical Geometry-II Three Marks Questions - by Question Bank Software - Sep 30, 2019 - View & Read

1)
Find the circumference and area of the circle x² +y² - 2x + 5y + 7 = 0
2)
Find the value of p so that 3x + 4y - p = 0 is a tangent to the circle x² +y²- 64 = 0.
3)
Find the condition for the line lx + my + n = 0 is tangent to the circle x² + y² = a²
4)
Find the area of th triangle found by the lines Joining the vertex of the parabola x² = -36y to the ends of the latus rectum.
5)
Find the equation of the ellipse whose e = \(\frac34\), foci ony-axis, centre at origin and passing through (6, 4).

12th Maths - Inverse Trigonometric Functions Three Marks Questions - by Question Bank Software - Sep 30, 2019 - View & Read

1)
Prove that \({ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) ={ tan }^{ -1 }\left( \frac { 27 }{ 11 } \right) \)
2)
Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)
3)
Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)
4)
Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)
5)
Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)

12th Maths - Theory of Equations Three Marks Questions - by Question Bank Software - Sep 30, 2019 - View & Read

1)
If α, β, and γ are the roots of the polynomial equation ax³+ bx²+ cx + d = 0, find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.
2)
If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.
3)
If the equations x²+ px + q = 0 and x²+ p'x + q' = 0 have a common root, show that it must be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).
4)
Solve the equation 9x³- 36x²+ 44x -16 = 0 if the roots form an arithmetic progression.
5)
Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.

12th Maths - Complex Numbers Three Marks Questions - by Question Bank Software - Sep 30, 2019 - View & Read

1)
Explain the falacy:
2)
Find the circle roots of -27.
3)
Find the principal value of -2i.
4)
Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.
5)
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

12th Maths - Application of Matrices and Determinants Three Marks Questions - by Question Bank Software - Sep 30, 2019 - View & Read

1)
If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A⁻¹ = A^T.
2)
If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = (adj A)A = |A|I₂.
3)
If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1
4)
Decrypt the received encoded message \(\left[ \begin{matrix} 2 & -3 \end{matrix} \right] \left[ \begin{matrix} 20 & 4 \end{matrix} \right] \) with the encryption matrix \(\left[ \begin{matrix} -1 & -1 \\ 2 & 1 \end{matrix} \right] \) and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26 to the letters A - Z respectively, and the number 0 to a blank space.
5)
Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 3 \\ \begin{matrix} 1 \\ 1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} -1 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -1 \\ \begin{matrix} 2 \\ \begin{matrix} 3 \\ 1 \end{matrix} \end{matrix} \end{matrix} \right] \)

12th Standard Maths - Complex Numbers Model Question Paper - by Meera - Namakkal - Sep 27, 2019 - View & Read

1)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
2)
The solution of the equation |z| - z = 1 + 2i is
3)
If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is
4)
If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is
5)
If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

12th Maths- Application of Matrices and Determinants Model Question Paper - by Meera - Namakkal - Sep 27, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
3)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is
4)
If \(\rho\) (A) = \(\rho\)([A| B]), then the system AX = B of linear equations is
5)
If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

12th Maths Quarterly Exam Question Paper 2019 - by Question Bank Software - Sep 21, 2019 - View & Read

12th Maths - Term 1 Model Question Paper - by Meera - Namakkal - Sep 21, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
3)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
4)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
5)
If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

12th Maths - Term 1 Five Mark Model Question Paper - by Question Bank Software - Sep 16, 2019 - View & Read

1)
If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.
2)
If ax² + bx + c is divided by x + 3, x − 5, and x − 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)
3)
Solve the system: x + y − 2z = 0, 2x − 3y + z = 0, 3x − 7y + 10z = 0, 6x − 9y + 10z = 0.
4)
Solve the following systems of linear equations by Cramer’s rule:
3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.
5)
Solve the equation z³+ 27 = 0

12th Maths - Applications of Vector Algebra Two Marks Questions - by Question Bank Software - Sep 16, 2019 - View & Read

1)
Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2, 3, 4) and parallel to the straight line \(\frac { x-1 }{ -4 } =\frac { y+3 }{ 5 } =\frac { 8-z }{ 6 } \)
2)
Find the vector and Cartesian equations of the plane passing through the point with position vector \(4\hat { i } +2\hat { j } -3\hat { k } \) and normal to vector \(2\hat { i } -\hat { j } +\hat { k } \)
3)
A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the coordinate axes is a constant. Show that the plane passes through a fixed point
4)
Find the vector and Cartesian equations of the plane passing through the point with position vector \(2\hat { i } +6\hat { j } +3\hat { k } \) and normal to the vector \(\hat { i } +3\hat { j } +5\hat { k } \)
5)
A plane passes through the point (−1, 1, 2) and the normal to the plane of magnitude \(3\sqrt { 3 } \) makes equal acute angles with the coordinate axes. Find the equation of the plane.

12th Maths - Two Dimensional Analytical Geometry II Two Marks Questions - by Question Bank Software - Sep 16, 2019 - View & Read

1)
Find the general equation of a circle with centre (-3, -4) and radius 3 units.
2)
Find the general equation of the circle whose diameter is the line segment joining the points (−4, −2)and (1, 1) is x²+y²+5x+3y+6=0
3)
Examine the position of the point (2, 3) with respect to the circle x²+ y²− 6x − 8y + 12 = 0.
4)
Find the equation of the circle with centre (2, -1) and passing through the point (3, 6) in standard form.
5)
Obtain the equation of the circle for which (3, 4) and (2, -7) are the ends of a diameter.

12th Maths - Inverse Trigonometric Functions Two Marks Questions - by Question Bank Software - Sep 16, 2019 - View & Read

1)
State the reason for cos^-1\([cos(-\frac{\pi}{6})]\neq \frac{\pi}{6}.\)
2)
Is cos^-1(-x) = \(\pi\)-cos⁻¹(x) true? Justify your answer.
3)
Find the principal value of cos^-1\((\frac{1}{2})\).
4)
Find the value of sec⁻¹\(\left( -\frac { 2\sqrt { 3 } }{ 3 } \right) \)
5)
If cot^-1\(\frac{1}{7}=\theta\), find the value of cos \(\theta\).

12th Maths - Theory of Equations Two Marks Questions - by Question Bank Software - Sep 16, 2019 - View & Read

1)
A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.
2)
Find the monic polynomial equation of minimum degree with real coefficients having 2 -\(\sqrt{3}\)i as a root.
3)
Find a polynomial equation of minimum degree with rational coefficients, having 2 + i as a root.
4)
Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.
5)
Show that the polynomial 9x⁹+ 2x⁵- x⁴- 7x²+ 2 has at least six imaginary roots.

12th Maths - Complex Numbers Two Marks Questions - by Question Bank Software - Sep 16, 2019 - View & Read

1)
If z = x + iy, find the following in rectangular form.
\(Re\left( \frac { 1 }{ z } \right) \)
2)
Represent the complex number −1−i
3)
Write the following in the rectangular form:
\(\cfrac { 10-5i }{ 6+2i } \)
4)
Find the square roots of −6+8i
5)
Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
\(\overline { z } =z^{ -1 }\)

12th Maths Unit 1 Application of Matrices and Determinants Two Marks Questions - by Question Bank Software - Sep 16, 2019 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
If adj A = \(\left[ \begin{matrix} -1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), find A⁻¹.
3)
Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right] \)
4)
Find the rank of the following matrices by minor method or show that the rank of matrix is 3
\(\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 2 \end{matrix} \end{matrix} \right] \)
5)
Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 6 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 0 \\ \begin{matrix} 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -9 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)

12th Maths Quarterly Model Question Paper - by Question Bank Software - Sep 13, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I₂, then B =
3)
If x^ay^b = e^m, x^cy^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,
4)
If A^T is the transpose of a square matrix A, then ___________
5)
If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--

TN 12th Standard Maths Official Model Question Paper 2019 - 2020 - by Question Bank Software - Sep 10, 2019 - View & Read

unit test - by QB365 School - Sep 04, 2019 - View & Read

12th Standard Maths Unit 6 Applications of Vector Algebra Book Back Questions - by Question Bank Software - Sep 04, 2019 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
3)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is
4)
If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are
5)
The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

12th Maths - Two Dimensional Analytical Geometry-II Book Back Questions - by Question Bank Software - Sep 03, 2019 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The radius of the circle 3x²+ by²+ 4bx − 6by + b² = 0 is
3)
If P(x, y) be any point on 16x²+ 25y²= 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁ + PF₂ is
4)
The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse is
5)
Tangents are drawn to the hyperbola \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

12th Standard Maths - Theory of Equations Book Back Questions - by Question Bank Software - Sep 02, 2019 - View & Read

1)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
2)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?
3)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
4)
The polynomial x³+ 2x + 3 has
5)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

12th Standard Maths - Inverse Trigonometric Functions Book Back Questions - by Question Bank Software - Sep 02, 2019 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is
3)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
4)
\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to
5)
If the function f(x) = sin^-1(x²- 3), then x belongs to

12th Standard Maths Unit 2 Complex Numbers Book Back Questions - by Question Bank Software - Aug 31, 2019 - View & Read

1)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to
2)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
3)
If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is
4)
If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂+ 4z₁z₃+ z₂z₃| = 12, then the value of |z₁+z₂+z₃| is
5)
The principal argument of \(\cfrac { 3 }{ -1+i } \) is

12th Standard Maths Unit 1 Application of Matrices and Determinants Book Back Questions - by Question Bank Software - Aug 30, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
4)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
5)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =

12th Standard Maths Unit 3 Theory of Equations One Mark Question and Answer - by Question Bank Software - Aug 29, 2019 - View & Read

1)
A zero of x³ + 64 is
2)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
3)
A polynomial equation in x of degree n always has
4)
If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________
5)
The quadratic equation whose roots are ∝ and β is ___________

12th Standard Maths Unit 2 Complex Numbers One Mark Question and Answer - by Question Bank Software - Aug 29, 2019 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
3)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
4)
.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________
5)
The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________

12th Standard Maths - Application of Matrices and Determinants One Mark Question and Answer - by Question Bank Software - Aug 28, 2019 - View & Read

1)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
2)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
3)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
4)
Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________
5)
If A^T is the transpose of a square matrix A, then ___________

12th Standard Physics Unit 6 Applications of Vector Algebra One Mark Question and Answer - by Question Bank Software - Aug 27, 2019 - View & Read

1)
If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to
2)
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
3)
If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } +\hat { j } \), \(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \), then the value of \(\lambda +\mu \) is
4)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to
5)
The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

12th Physics Chapter 5 Two Dimensional Analytical Geometry-II One Mark Question and Answer - by Question Bank Software - Aug 27, 2019 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The radius of the circle 3x²+ by²+ 4bx − 6by + b² = 0 is
3)
The centre of the circle inscribed in a square formed by the lines x²− 8x − 12 = 0 and y²− 14y + 45 = 0 is
4)
Equation of tangent at (-4, -4) on x² = -4y is _____________
5)
y²- 2x - 2y + 5 = 0 is a _________

12th Standard Physics Chapter 4 Inverse Trigonometric Functions One Mark Question and Answer - by Question Bank Software - Aug 27, 2019 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
\(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}+\sec ^{-1} \frac{5}{3}-\operatorname{cosec}^{-1} \frac{13}{12}\) is equal to
3)
If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is
4)
If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________
5)
The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

12th Physics Unit 2 Theory of Equations One Mark Question with Answer Key - by Question Bank Software - Aug 26, 2019 - View & Read

1)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
2)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
3)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
4)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is
5)
If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________

12th Maths Chapter 2 Complex Numbers One Mark Question Paper - by Question Bank Software - Aug 24, 2019 - View & Read

1)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
2)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
3)
If |z - 2 + i | ≤ 2, then the greatest value of |z| is
4)
If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is
5)
The solution of the equation |z| - z = 1 + 2i is

Unit test 12th Standard Maths New syllabus - by QB365 School - Aug 23, 2019 - View & Read

12th Maths Chapter 1 Application of Matrices and Determinants One Mark Questions - by Question Bank Software - Aug 23, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
4)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
5)
If A, B and C are invertible matrices of some order, then which one of the following is not true?

12th Maths Quarterly Exam Model Two Marks Question Paper - by Question Bank Software - Aug 20, 2019 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
Reduce the matrix \(\left[ \begin{matrix} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{matrix} \right] \) to a row-echelon form.
3)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)
4)
Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form
5)
Represent the complex number −1−i

12th Maths Unit 6 Applications of Vector Algebra Model Question Paper - by Question Bank Software - Aug 12, 2019 - View & Read

1)
If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then
2)
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
3)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is
4)
The angle between the lines \(\frac{x-2}{3}=\frac{y+1}{-2}, z=2 \text { and } \frac{x-1}{1}=\frac{2 y+3}{3}=\frac{z+5}{2}\) is
5)
Distance from the origin to the plane 3x − 6y + 2z + 7 = 0 is

12th Standard Maths Quarterly Exam Model One Mark Question Paper - by Question Bank Software - Aug 05, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
3)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
4)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =
5)
If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is

Plus 2 Maths Chapter 5 Two Dimensional Analytical Geometry - II Model Questions - by Question Bank Software - Aug 03, 2019 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The centre of the circle inscribed in a square formed by the lines x²− 8x − 12 = 0 and y²− 14y + 45 = 0 is
3)
If x + y = k is a normal to the parabola y² = 12x, then the value of k is
4)
Equation of tangent at (-4, -4) on x² = -4y is _____________
5)
y²- 2x - 2y + 5 = 0 is a _________

12th Standard Maths First Mid Term Model Question Paper - by Question Bank Software - Aug 01, 2019 - View & Read

1)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
2)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
3)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
4)
The value of (1+i)⁴ + (1-i)⁴ is __________
5)
The value of \(\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } } \) is __________

11th Standard Mathematics Chapter 4 Inverse Trigonometric Functions Important Question Paper - by Question Bank Software - Jul 30, 2019 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to
3)
\(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}+\sec ^{-1} \frac{5}{3}-\operatorname{cosec}^{-1} \frac{13}{12}\) is equal to
4)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________
5)
The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi is ___________

12th Standard Maths Chapter 3 Theory of Equations Important Question Paper - by Question Bank Software - Jul 26, 2019 - View & Read

1)
A zero of x³ + 64 is
2)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
3)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
4)
The polynomial x³+ 2x + 3 has
5)
If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots

Model MID-TERM - by MUTHU M - Jul 24, 2019 - View & Read

12th Maths Unit 2 Important Question Paper - by Question Bank Software - Jul 24, 2019 - View & Read

1)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
2)
If z is a non zero complex number, such that 2iz² =\(\bar { z } \) then |z| is
3)
z₁, z₂ and z₃ are complex number such that z₁+ z₂+ z₃= 0 and |z₁| = |z₂| = |z₃| = 1 then z₁²+ z₂²+ z₃³ is
4)
If z₁, z₂, z₃ are the vertices of a parallelogram, then the fourth vertex z₄ opposite to z₂ is _____
5)
If x_r= \(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x₁, x_2,x₃ ... x_∞ is _________

Slip Test Unit 3 (A2) - by MUTHU M - Jul 23, 2019 - View & Read

1)
Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.
2)
If α, β, γ and \(\delta\) are the roots of the polynomial equation 2x⁴+ 5x³− 7x²+ 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).
3)
Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.
4)
Solve: (2x-1) (x+3) (x-2) (2x+3)+20 = 0
5)
Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.

slip test - by MUTHU M - Jul 23, 2019 - View & Read

1)
Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.
2)
If the equations x²+ px + q = 0 and x²+ p'x + q' = 0 have a common root, show that it must be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).
3)
A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.
4)
If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.
5)
If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

Weekly test-1:JUNE2019 - by MUTHU M - Jul 23, 2019 - View & Read

1)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
2)
If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =
3)
If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I₂, then B =
4)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________
5)
In a square matrix the minor M_ij and the co-factor A_ij of and element a_ij are related by _____

12th Maths - Unit 1 Full Important Question Paper - by Question Bank Software - Jul 18, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
4)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
5)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =

frequently asked two marks in twelfth standard maths english medium - by Mythily - Jun 28, 2019 - View & Read

1)
For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.
2)
For the matrix A, if A³ = I, then find A^-1.
3)
If A is a square matrix such that A³ = I, then prove that A is non-singular.
4)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
5)
Find the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).

+2 english medium creative multiple choice questions in maths chapter one - by Mythily - Jun 28, 2019 - View & Read

1)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
2)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
3)
Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________
4)
If A^T is the transpose of a square matrix A, then ___________
5)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________

Important one mark questions 12th maths english medium chapter one - by Mythily - Jun 28, 2019 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If A is a 3 \(\times\) 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =
3)
If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =
4)
If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =
5)
If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A =

UNIT TEST - 1 - by Palanivel - May 16, 2019 - View & Read

1)
If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]^-1 = F(-\(\alpha\)).
2)
If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.
3)
If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A⁻¹ = A^T.
4)
If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1
5)
If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

x	1	2	3	4
f (x)	\(\cfrac { 1 }{ 12 } \)	\(\cfrac { 5 }{ 12 } \)	\(\cfrac { 5 }{ 12 } \)	\(\cfrac { 1 }{ 12 } \)

p	q	(p ∧ q) ∨ (¬q)
T	T	(a)
T	F	(b)
F	T	(c)
F	F	(d)

x	1	2	3	4	5
f(x)	k²	2k²	3k²	2k	3k

x	1	2	3	4	5
f(x)	k²	2k²	3k²	2k	3k

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