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

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

• 1)

  Find the area bounded by the curve y=xe^x and y=xe^-x and 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 - 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_e 2.

• 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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^-1 z = \(\pi\), 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 - 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 - 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 - 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 - 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 - 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 - 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^-1 x + 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 ^_o g)(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 - 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 - 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 - 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 - 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) = z₁ z₂ + z₁ z₃

• 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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