If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

Solve : \({{x^2-4}\over{x^2-2x-15}}\le0\)

The shaded region in the adjoining diagram represents.

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

The relation R defined on a set A= {0,-1, 1, 2} by xRy if |x²+y²| ≤ 2, then which one of the following is true?

If g(x) = (x² + 2x + 3) f(x) and f(0) = 5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\), then g'(0) is

The derivative of f(x) = x |x| at x = −3 is

 Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } \) then \(\frac { dy }{ dx } \) is _____

Choose the correct or the most suitable answer from the given four alternatives.
If \(f\left( x \right) =x+1\), then \(\frac { d }{ dx } ({ f }_{ 0 }f\left( x \right) )\) is _________

Choose the correct or the most suitable answer from the given four alternatives.
For the curve \(\sqrt { x } +\sqrt { y } =1,\quad \frac { dy }{ dx } at\left( \frac { 1 }{ 4 } ,\frac { 1 }{ 4 } \right) is\) _________

If a_ij = \({1\over2}(3i-2j)\) and A = [a_ij]_2x2 is

If A = \(\begin{bmatrix}\lambda & 1 \\ -1 & -\lambda \end{bmatrix}\), then for what value of \(\lambda\), A² = O?

The value of x, for which the matrix A = \(\begin{bmatrix} e^{x-2}& e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}\) is singular

If \(\triangle\) = \(\begin{vmatrix} a&b &c \\ x & y & z \\ p &q &r \end{vmatrix}\), then \(\begin{vmatrix} ka&kb &kc \\ kx & ky & kz \\k p &kq &kr \end{vmatrix}\) is

If a \(\neq\) b, b, c satisfy \(\begin{vmatrix} a&2b &2c \\3 & b & c \\ 4 & a & b \end{vmatrix}=0,\) then abc =

What must be the matrix X, if 2x +\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

If the points (x,−2), (5, 2), (8, 8) are collinear, then x is equal to

If the square of the matrix \(\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is the unit matrix of order 2, then \(\alpha ,\beta \) and \(\gamma\) should satisfy the relation.

If A is skew-symmetric of order n and C is a column matrix of order n \(\times\) 1, then C^T AC is

If A(B + C) = AB + AC where A, B, C are matrices of the same order than the property applied is matrix multipication is _______ .

The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-2\hat{j}+\hat{k}\) is

The vectors \(\overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c},\overrightarrow{c}-\overrightarrow{a}\) are

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

The vectors from origin to the points A and B are \(2\hat { i } -3\hat { j } +2\hat { k } \) and \(2\hat { i } +3\hat { j } +\hat { k } \) respectively, then the area of \(\Delta\)OAB is equal to

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to

If ABCD is a parallelogram, then \(\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}\) is equal to

If \(\overrightarrow{a},\overrightarrow{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

If \(\overrightarrow{r}={9\overrightarrow{a}+7\overrightarrow{b}\over16}\), then the point P whose position vector \(\overrightarrow{r}\) divides the line joining the points with position vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) in the ratio

\(lim_{x\rightarrow\infty}{sin \ x \over x} \)

\(lim_{x\rightarrow0}{\sqrt{1-cos 2x}\over x} \)

If f(x) = x(-1)^{\(\left\lfloor 1\over x \right\rfloor \)}, \(x\le0\), then the value of \(lim_{x\rightarrow 0}f(x)\) is equal to

Let the function f be defined by \(f(x)=\left\{\begin{array}{ll} 3 x & 0 \leq x \leq 1 \\ -3 x+5 & 1<x \leq 2 \end{array},\right. \text { then }\)

The value of \(lim_{x \rightarrow 0}{sin x\over \sqrt{x^2}}\) is

\(lim_{x\rightarrow {\pi/2}}{2x-\pi\over cosx} \)

\(lim_{x \rightarrow \infty}{\sqrt{x^2-1}\over 2x+1}=\)

\(lim_{x \rightarrow 3}\left\lfloor x \right\rfloor =\)

If f : \(R \rightarrow R\) is defined by f(x)=\(\left\lfloor x-3 \right\rfloor +|x-4|\) for \(x \in R\), then \(lim_{x\rightarrow 3^-}f(x)\) is equal to

\(lim_{n \rightarrow \infty}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

\(\frac{d}{d x}\left(\frac{2}{\pi} \sin x^{\circ}\right)\) is

If y = mx + c and f(0) =\(f '(0)=1\), then f(2) is

\({d\over dx}(e^{x+5log \ x})\) is

The differential coefficient of log₁₀ x with respect to log_x10 is

\(\text { If } f(x)= \begin{cases}x-5 & \text { if } x \leq 1 \\ 4 x^2-9 & \text { if } 1<x<2 \\ 3 x+4 & \text { if } x \geq 2\end{cases}\), then the right hand derivative of f(x) at x = 2 is

If \(\int f(x) d x=g(x)+c\), then \(\int f(x) g^{\prime}(x) d x\)

If \(\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x\) is

\(\int \sin ^3 x d x\) is

\(\int \frac{e^x\left(x^2 \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^2+1} d x\) is

\(\int \frac{\sec ^2 x}{\tan ^2 x-1} d x\) is

If \(\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c\), then the value of k is

\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\) is

\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x\) is

\(\int \frac{x^2+\cos ^2 x}{x^2+1} \operatorname{cosec}^2 x d x\) is

\(\int e^{-4 x} \cos x d x\) is

Let A and B be two events such that \(P(\overline{A\cup B})={1\over6}, P(A\cap B)={1\over4}\) and \({P(\overline{A})}={1\over4}\). Then the events A and B are

A man has 3 fifty rupee notes, 4 hundred rupees notes, and 6 five hundred rupees notes in his pocket. If 2 notes are taken at random, what are the odds in favour of both notes being of hundred rupee denomination?

If A and B are two events such that A ⊂ B and P(B)\(\neq o\), then which of the following is correct?

If X and Y be two events such that P(X/Y) = \({1\over2},P(Y/X)={1\over3}\) and \(P(X\cap Y)={1\over6}\), then P(X\(\cup\)Y) is

If two events A and B are such that \(P(\overline{A})={3\over10}\) and \(P(A \cap \overline{B})={1\over2},\) then \(P(A\cap B)\) is

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

Assertion (A) : cos x = \(\frac{-1}{2}\) and 0\(\frac{2\pi}{3},\frac{4\pi}{3}\).
Reason (R) : cos is negative in the first and fourth quadrant only.

The sum up to n terms of the series \(\frac { 1 }{ \sqrt { 1 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 3 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 7 } } +\)....is 

The sum of the series C₀²- C₁² + C₂² .....+ (- 1)ⁿC²_n where n is an even integer is ______________

The range of the function \(f(x) = \left| \left\lfloor x \right\rfloor - x \right| ,x \in R\)  is 

The condition that the equation ax² + bx + c = 0 may have one root is the double the other is ___________

If \(\alpha\) and \(\beta\) are two values of θ obtained from the equation a cos θ + b sin θ = c then the value of \(tan(\frac{\alpha+\beta}{2})\) is _______________

a polygon has 44 diagonals, then the number of its sides are _________

\(\frac{1}{q+r},\frac{1}{r+p},\frac{1}{p+q}\) are in A.P., then ______________

If 3 is the logarithm of 343, then the base is

If the arcs of same lengths in two circles sustend central angles 30° and 40° find the ratio of their radii _______________

The value of cos 20°- sin 20° is _______________

If ⁿC₄,ⁿC₅,ⁿC₆ are in AP the value of n can be

Each of five questions is a multiple-choice test has 4 possible answers. The number of different sets of possible answers is  _________

A number is selected from the set {1,2,3,...,20}. The probability that the selected number is divisible by 3 or 4 is

A bag contains 5 white and 3 black balls. Five balls are drawn successively without replacement. The probability that they are alternately of different colours is

A bag contains 6 green, 2 white, and 7 black balls. If two balls are drawn simultaneously, then the probability that both are different colours is

If two events A and B are independent such that P(A) = 0.35 and \(P(A\cup B)=0.6\), then P(B) is

In a certain college 4% of the boys and 1% of the girls are taller than 1.8 meter. Further 60% of the students are girls. If a student is selected at random and is taller than 1.8 meters, then the probability that the student is a girl is

The function f:R➝R be defined by f(x) = sinx + cosx is

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

\(\sqrt [ 4 ]{ 11 } \) is equal to ___________

If tan α and tan β are the roots of x² + ax + b = 0; then \(\frac { sin(\alpha +\beta ) }{ sin\alpha sin\beta } \) is equal to

If tan A = \(\frac { a }{ a+1 } \) and B = \(\frac { 1 }{ 2a+1 } \) then the value of A + B is ___________

If \(x={1\over 2+\sqrt{3}}\) then the value of x³ - x² - 11x + 3 is 

If sin α + cos α = b, then sin 2α is equal to

If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

The number of positive integral solution of \(x\times y\times z=30\) is  _________

Which one of the following statements in false?

The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

\(\sqrt [ 4 ]{ 11 } \) is equal to ___________

Solve 3x² + 5x - 2≤0

The value of log 1 is

If cos x = \(\frac { -1 }{ 2 } \) \(0 < x < 2\pi\) and, then the solutions are _______________

Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

\(n(p(A))=512,n(p(B))=32,n(A\cup B)=16,\) find \(n(A\cap B) \)  ___________

If A and B are any two finite sets having m and n elements respectively then the cardinality of the power set of A \(\times\) B is ___________

The value of log₁₀⁸ + log₁₀⁵- log₁₀⁴ = ___________

Everybody in a room shakes hands with everybody else. The total number of handshakes is 91. The total number of persons in the room is _________

The function f:R➝R be defined by f(x) = sinx + cosx is

The number of reflective relations one set containing n elements is __________

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

2tan^-1\(\left( \frac { 1 }{ 5 } \right) \) is equal to _______________

The number of ways in which we can arrange 4 letters of the word "MATHEMATICS" is given by  _________

If n((A \(\times\) B) ∩(A \(\times\) C)) = 8 and n(B ∩ C) = 2, then n(A) is

Domain of the function \(y={x-1\over x+1}\) is __________

Which one of the following is false?

Find a so that the sum and product of the roots of the equation 2x²+ (a - 3) x + 3a - 5 = 0 are equal is

If \(\alpha\) and \(\beta\) are the roots of 2x² - 3x - 4 = 0 find the value of \(\alpha^2+\beta^2\)

The function f(x) = log (x + \(\sqrt{x^2+1}\)) is ___________

If a and b are the roots of the equation x²- kx + 16 = 0 and a²+ b² = 32, then the value of k is

The number of real solution of |2x - x²- 3| = 1 is ___________

If ⁿP_r=k x ^n-1P_r-1 what is k:

Choose the incorrect pair:

If two sets A and B have 17 elements in common, then the number of elements common to the set A \(\times\)B and B \(\times\)A is

Which one of the following statements is false? The graph of the function \(f(x)={1\over x}\)

If |x + 3| ≥ 10 then ___________

The quadratic equation whose roots are tan 75° and cot 75° is _______________

If 10ⁿ + 3 \(\times\) 4ⁿ⁺²+\(\lambda \) is divisible by 9 for all n \(\in \)N, then the least positive integral value of \(\lambda \) is _________

If A = {(x,y) : y = sin x, x ∈ R} and B = {(x,y) : y = cos x, x ∈ R} then A∩B contains

Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

Domain of the function \(y={x-1\over x+1}\) is __________

The number of solution of x² + |x - 1| = 1 is

Choose the incorrect statement

The function f:R➝R is defined by f(x)=\(\frac { \left( { x }^{ 2 }+cosx \right) \left( 1+{ x }^{ 4 } \right) }{ \left( x-sinx \right) \left( 2x-{ x }^{ 3 } \right) } +{ e }^{ -\left| x \right| }\) is

Let A = {-2, -1, 0, 1, 2} and f : A ⟶ Z be given by f(x) = x²- 2x - 3 then preimage of 5 is ___________

If sin(45 ° + 10°) - sin(45° -10°) = \(\sqrt{2}\)sin x then x is ___________ 

The numerical value of tan^-11 + tan^-12 + tan^-13 = _______________

The n^th term of the sequence \(\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 8 } ,\frac { 15 }{ 6 } \),......is

If A = {(x,y) : y = e^x, x∈R} and B = {(x,y) : y = e^-x, x ∈ R} then n(A∩B) is

If the roots of x²-bx + c = 0 are two consecutive integer,then b²- 4c is ___________

Logarithm of 144 to the base 2\(\sqrt{3}\) is ___________

The value of \(\sqrt [ 4 ]{ { (-2) }^{ 4 } } =\) _______.

cos 6x - cos 8x = _______________

If A = {(x,y) : y = e^x, x∈R} and B = {(x,y) : y = e^-x, x ∈ R} then n(A∩B) is

If A = {x / x is an integer, x² \(\le\) 4} then elements of A are ___________

Solve \(\sqrt{7+6x-x^2}=x+1\)

The value of log 1 is

If \(\Sigma n=210\) then \(\Sigma { n }^{ 2 }\)= ______________

The function f(x) = log (x + \(\sqrt{x^2+1}\)) is ___________

The value of log₁₀⁸ + log₁₀⁵- log₁₀⁴ = ___________

For x≥2, |x-2|=

\(\left(1+\frac{1}{\lfloor2}+\frac{1}{\lfloor4}+\frac{1}{\lfloor6}+...\right)^2-\left(1+\frac{1}{\lfloor3}+\frac{1}{\lfloor5}+\frac{1}{\lfloor7}+...\right)^2=\)______________

Find the nearest point on the line 3x + y = 10 from the origin is ______________

Let A = {-2, -1, 0, 1, 2} and f : A ⟶ Z be given by f(x) = x²- 2x - 3 then preimage of 5 is ___________

If a and b are the real roots of the equation x²- kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

Zero of the polynomial p(x) = x² - 4x + 4

In a triangle ABC, sin²A + sin²B + sin²C = 2, then the triangle is

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

The range of the function \({1\over 1-2sinx}\) is

cos p = \(\frac { 1 }{ 7 } \) and cos Q = \(\frac { 13 }{ 14 } \) where P, Q are angles, then P-Q is _______________

The coefficient of a⁵ in the expansion of (3a + 5b)⁵ is ______________

If(1, 3) (2,1) (9, 4) are collinear then a is ______________

The function \(f\left( x \right) =\tan { x } \) is discontinuous on the set

\(n(p(A))=512,n(p(B))=32,n(A\cup B)=16,\) find \(n(A\cap B) \)  ___________

If 3 is the logarithm of 343, then the base is

\(\frac { cos3x }{ 2cos2x-1 } \) is _______________

The number of positive integral solution of \(x\times y\times z=30\) is  _________

The sum of the digits in the unit's place of all the 4- digit numbers formed by 3, 4, 5 and 6, without repetition, is _______.

If \(f:R\rightarrow R\) is defined by \(f(x)=2x-3\) __________

The triangle of maximum area with constant perimeter 12m

tan 70°- tan 20°= _____________ 

The product of first n odd natural numbers equals

The middle term in the expansion of  is \((x- \frac{2}{x})^{12}\) is ______________

The HM of two positive numbers whose AM and GM are 16, 8 respectively is

The sum up to n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\).....is

The sum of an infinite GP is 18. If the first term is 6, the common ratio is

The value of \(\frac { 1 }{ 2! } +\frac { 1 }{ 4! } +\frac { 1 }{ 6! } +....is\)

The first and last term of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be ______________

The equation of the locus of the point whose distance from y-axis is half the distance from origin is

The intercepts of the perpendicular bisector of the line segment joining (1, 2) and (3, 4) with coordinate axes are

The point on the line 2x- 3y = 5 is equidistance from (1, 2) and (3, 4) is

The length of \(\bot\) from the origin to the line \(\frac{x}{3}-\frac{y}{4}=1,\) is 

If one of the lines given by 6x² - xy + 4cy² = 0 is 3x + 4y = 0, then c equals to

Which of the following equation is the locus of (at², 2at)

The slope of the line which makes an angle 45^o with the line 3x- y = -5 are:

The coordinates of the four vertices of a quadrilateral are (-2, 4), (-1, 2), (1, 2) and (2, 4) taken in order. The equation of the line passing through the vertex (-1, 2) and dividing the quadrilateral in the equal areas is

Equation of the straight line perpendicular to the line x - y + 5 = 0, through the point of intersection the y-axis and the given line

The line (p + 2q)x + (p - 3q)y = p - q for different values of p and q passes through the point

The HM of two positive numbers whose AM and GM are 16, 8 respectively is

The sum of an infinite GP is 18. If the first term is 6, the common ratio is

If the sum of n terms of an A. P. be 3n² - n and its common difference is 6, then its first term is ______________

If in an infinite G. P. first term is equal to 10 times the sum of all successive terms, then its common ratio is ______________

If \(\Sigma n=210\) then \(\Sigma { n }^{ 2 }\)= ______________

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is

The number of 10 digit number that can be written by using the digits 2 and 3 is

The product of first n odd natural numbers equals

The number of constant functions from a set containing m elements to a set containing n elements is

If |x+2| \(\le\) 9, then x belongs to

\(\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } } \)=

The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

If a, 8, b are in AP, a, 4, b are in GP, and if a, x, b are in HP then x is

The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

Given that x, y and b are real numbers x < y, b > 0, then

If cos 28⁰+ sin 28⁰ = k³, then cos 17⁰ is equal to

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

The sequence \(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } }, \frac { 1 }{ \sqrt { 3 } +2\sqrt { 2 } },...... \)form an 

Let X = {1, 2, 3, 4}, Y = {a, b, c, d} and f = {(1, a), (4, b), (2, c), (3, d), (2, d)}. Then f is

If \(\frac { |x-2| }{ x-2 } \ge 0\), then x belongs to

\(\left( 1+cos\frac { \pi }{ 8 } \right) \left( 1+cos\frac { 3\pi }{ 8 } \right) \left( 1+cos\frac { 5\pi }{ 8 } \right) \left( 1+cos\frac { 7\pi }{ 8 } \right) \) =

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

The HM of two positive numbers whose AM and GM are 16, 8 respectively is

The inverse of f(x) = \(\begin{cases} x\quad if\quad x<1 \\ { x }^{ 2 }\quad if\quad 1\le x\le 4 \\ 8\sqrt { x } \quad if\quad x>4 \end{cases}\) is

If 8 and 2 are the roots of x²+ ax + c = 0 and 3, 3 are the roots of x² + dx + b = 0; then the roots of the equation x²+ ax + b = 0 are

The product of r consecutive positive integers is divisible by

The sum up to n terms of the series \(\frac { 1 }{ \sqrt { 1 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 3 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 7 } } +\)....is 

If a is the arithmetic mean and g is the geometric mean of two numbers, then

Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

Let f : Z➝Z be given by f(x) = \(\begin{cases} \frac { x }{ 2 } \quad if\quad x\quad is\quad even \\ 0\quad if\quad x\quad is\quad odd \end{cases}\). Then f is __________

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

If cos pፀ + cos qፀ = 0 and if p ≠ q, then ፀ is equal to (n is any integer)

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

\(\frac { sin(A-B) }{ cosAcosB } +\frac { sin(B-C) }{ cosBcosC } +\frac { sin(C-A) }{ cosCcosA } \) is

The number of five digit telephone numbers having at least one of their digits repeated is

Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter 4 + 2\(\sqrt{2}\) is

If A =\(\begin{bmatrix} 1& 2 &2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}\) is a matrix satisfying the equation AA^T = 9I, where I is 3 \(\times\) 3 identity matrix, then the ordered pair (a, b) is equal to

Which one of the following is a finite set?

The value of \(\sqrt [ 4 ]{ { (-2) }^{ 4 } } =\) _______.

The value of sin²\(\frac { 5\pi }{ 12 } -sin^{ 2 }\frac { \pi }{ 12 } \) is ___________

If ⁿC₁₀ = ⁿC₆, then ⁿC₂ =  _________

The value of \({ 9 }^{ \frac { 1 }{ 3 } }\) ,\({ 9 }^{ \frac { 1 }{ 9 } }\)\({ 9 }^{ \frac { 1 }{ 27}}\),\(\infty \) is ______________

The shaded region in the adjoining diagram represents.

The number of reflective relations one set containing n elements is __________

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

If cosec x + cot x = \(\frac { 11 }{ 2 } \) then tan x = ___________

Among the players 5 are bowlers. In how many ways a team of 11 may be formed with atleast 4 bowlers?

If \(f:[-2,2]\rightarrow A\) is given by f(x) = 3³ then f is onto, if A is ___________

The number of ways in which we can arrange 4 letters of the word "MATHEMATICS" is given by  _________

The coefficient of a⁵ in the expansion of (3a + 5b)⁵ is ______________

The length of perpendicular from the origin to a line is 12 and the line makes an angle of 120° with the positive direction of y-axis. then the equation of line is ______________

If co-ordinate axes are the angle bisectors of the pair of lines ax²+ 2hxy + by² = 0 then ______________

The number of reflective relations one set containing n elements is __________

For the below figure of ax² + bx + c = 0

Among the players 5 are bowlers. In how many ways a team of 11 may be formed with atleast 4 bowlers?

The Co-efficient of x³ in \(\sqrt { \frac { 1-x }{ 1+x } } ,\left| x \right| <1\ is\ \)______________

\(\frac{1}{q+r},\frac{1}{r+p},\frac{1}{p+q}\) are in A.P., then ______________

Let S = (1, 2, 3), R be (1, 1) (1, 2) (2, 2) (1, 3) (3, 1), what are the elements to-be included to make R reflexive ___________

(x²-2x+2)(x²+2x+2) are the factors of the polynomial ___________

The product of r consecutive positive integers is divisible by _________

The middle term in the expansion of  is \((x- \frac{2}{x})^{12}\) is ______________

If \(\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix}=\begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}\), then the value of x+y is _________ .

The number of relations from a set containing 4 elements to a set containing 3 elements is:

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

2 sin 5x cos x _______________

If 15C_3r = 15 C_r+3 , then r is equal to _________

If \(\Sigma n=210\) then \(\Sigma { n }^{ 2 }\)= ______________

If tan x = \(\frac { -1 }{ \sqrt { 5 } } \) and x lies in the IV quadrant, then the value of cos x is ___________

cos 35⁰ + cos 85⁰ + cos 155⁰ = _______________

The value of sin 20° sin 40° sin 60° sin18° is _______________

If (A + B) = \(\frac{\pi}{4}\), (cot A - 1) (cot B - 1) = _______________

The value of tan^-1 (1) + cos^-1(\(\frac{-1}{2}\)) + sin^-1(\(\frac{-1}{2}\)) _______________

Which of the following functions from z to itself are bijections (one-one and onto)?

The domain of the function \(f(x)=\sqrt{4-\sqrt{4-\sqrt{4-x^2}}}\)

The rationalising factor of \(\frac { 5 }{ \sqrt [ 3 ]{ 3 } } \) is

The condition that the equation ax² + bx + c = 0 may have one root is the double the other is ___________

Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is _________

Let X = {a, b,c},y = (1, 2, 3) then \(f:x\rightarrow y\) given by (a, 1) (b, 1) (c, 1) is called ___________

Which one of the following is false?

The logarithmic form of 5² = 25 is ___________

If 15C_3r = 15 C_r+3 , then r is equal to _________

ⁿC_r + ²ⁿC_r-1 + ⁿC_r-2 =  _________

The value of a when x³- 2x²+ 3x + a is divided by (x - 1), the remainder is 1, is ___________

If ⁿC_r-1 = 36, ⁿC_r = 84 and ⁿC_r+1 = 126 then r = _________

2^1/4 4^1/8 8^1/16 16^1/32 . . . = ______________

If the co-ordinates of a variable point p be \((t+\frac{1}{t},t-\frac{1}{t})\) where t is the parameter then the locus of p ______________

If A is a matrix 3 \(\times\) 3, then \({ { (A }^{ 2 }) }^{ -1 }\) =____________

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

The relation R defined on a set A= {0,-1, 1, 2} by xRy if |x²+y²| ≤ 2, then which one of the following is true?

The domain of the function \(f(x)=\sqrt{log_{10}{3-x\over x}}\)is

The domain and range of the function \(f(x)={|x-4|\over x-4}\)

The value of \({ log }_{ 3 }\frac { 1 }{ 81 } \) is

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

If f(x) = |x - 2| + |x + 2|, x ∈ R, then

Let S = (1, 2, 3), R be (1, 1) (1, 2) (2, 2) (1, 3) (3, 1), what are the elements to-be included to make R reflexive ___________

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

If A and B are two matrices such that A + B and AB are both defined, then

If the square of the matrix \(\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is the unit matrix of order 2, then \(\alpha ,\beta \) and \(\gamma\) should satisfy the relation.

If A + I =\(\begin{bmatrix} 3& -2 \\ 4 & 1 \end{bmatrix}\), then (A + I )(A - I) is equal to

If A and B are square matrices of order 3 and |A| = 5, |B| = 3 then |3 AB| is _____________

One of the diagonals of parallelogram ABCD with \(\overrightarrow{a}\) and \(\overrightarrow{b}\) as adjacent sides is \(\overrightarrow{a}+\overrightarrow{b}\) The other diagonal \(\overrightarrow{BD}\) is

If \(\begin{vmatrix}2a & x_1 &y_1 \\ 2b & x_2 & y_2 \\ 2c & x_3 &y_3 \end{vmatrix}={abc\over 2}\neq 0,\) then the area of the triangle whose vertices are \(\begin{pmatrix} {x_1\over a}, {y_1\over a} \end{pmatrix}\), \(\begin{pmatrix} {x_2\over b}, {y_2\over b} \end{pmatrix}\), \(\begin{pmatrix} {x_3\over c}, {y_3\over c} \end{pmatrix}\) is

The value of the determinant of A = \(\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is\)

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

Let f :\(R \rightarrow R\) be defined by \(f(x)= \begin{cases}x & x \text { is irrational } \\ 1-x & x \text { is rational }\end{cases}\)  then f is

\(\lim _{ x\rightarrow 1 }{ \frac { { x }^{ m }-1 }{ { x }^{ n }-1 } } is\)

The number of constant functions from a set containing m elements to a set containing n elements is

If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

Which one of the following is a finite set?

Given A = {5,6,7,8}. Which one of the following is incorrect?

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

The shaded region in the adjoining diagram represents.

If f(x) = |x - 2| + |x + 2|, x ∈ R, then

If two sets A and B have 17 elements in common, then the number of elements common to the set A \(\times\)B and B \(\times\)A is

Let f : Z➝Z be given by f(x) = \(\begin{cases} \frac { x }{ 2 } \quad if\quad x\quad is\quad even \\ 0\quad if\quad x\quad is\quad odd \end{cases}\). Then f is __________

If |x+2| \(\le\) 9, then x belongs to

The value of log_a b log_b c log_c a is

Find a so that the sum and product of the roots of the equation 2x²+ (a - 3) x + 3a - 5 = 0 are equal is

If 8 and 2 are the roots of x²+ ax + c = 0 and 3, 3 are the roots of x² + dx + b = 0; then the roots of the equation x²+ ax + b = 0 are

If  \(\frac { kx }{ (x+2)(x-1) } =\frac { 2 }{ x+2 } +\frac { 1 }{ x-1 } \), then the value of k is

Given that x, y and b are real numbers x < y, b > 0, then

If 3 is the logarithm of 343, then the base is

If a and b are the real roots of the equation x²- kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

If  \(\frac { 1-2x }{ 3+2x-{ x }^{ 2 } } =\frac { A }{ 3-x } +\frac { B }{ x+1 } \), then the value of A + B is

If - 3x + 17 < -13 then ___________

\(\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } } \)=

The maximum value of 4sin²x + 3 cos²x + \(sin\frac { x }{ 2 } +cos\frac { x }{ 2 } \) is

cos1⁰ + cos2⁰ + cos3⁰ +: : : + cos179⁰ =

If tan α and tan β are the roots of x² + ax + b = 0; then \(\frac { sin(\alpha +\beta ) }{ sin\alpha sin\beta } \) is equal to

If sin α + cos α = b, then sin 2α is equal to

If cos 28⁰+ sin 28⁰ = k³, then cos 17⁰ is equal to

If tan40⁰ = λ, then \(\frac { tan{ 140 }^{ 0 }-tan{ 130 }^{ 0 } }{ 1+tan{ 140 }^{ 0 }.tan{ 130 }^{ 0 } } \) =

Let f_k(x) = \(\frac { 1 }{ k } \)[sin^kx + cos^kx] where x\(\in \)R and k ≥ 1. Then f₄(x) - f₆(x) =

In a triangle ABC, sin²A + sin²B + sin²C = 2, then the triangle is

A wheel is spinning at 2 radians/second. How many seconds will it take to make 10 complete rotations?

The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is

The product of r consecutive positive integers is divisible by _________

By taking suitable sets A, B, C, verify the following results:
C-(B-A) = (C\(\cap \) A) \(\cup \) (C\(\cap \)B')

Discuss the following relations for reflexivity, symmetricity and transitivity:
Let A be the set consisting of all the members of a family. The relation R defined by "aRb if a is not a sister of b".

Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

Draw the curves of
(i) y = x² + 1
(ii) Y = (x + 1)² by using the graph of curve y = x.

A simple cipher takes a number and codes it, using the function f(x) = 3x - 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x(by drawing the lines)

Graph the function f(x) = x³ and \(g(x)=\sqrt[3]x\) on the same co-ordinate plane. Find f o g and graph it on the plane as well. Explain your results.

Write the values of f at -3, 5, 2, -1, 0 if
\(f(x)=\begin{cases} x^2+x-5\quad if\ x \in(-\infty, 0) \\x^2+3x-2\quad if\ x\in(3,\infty) \\x^2\quad \quad \quad \quad \quad if\ x\ \in(0,2) \\x^2-3 \quad \quad \quad otherwise \end{cases}\)

If a² = by + cz, b² = cz + ax and c² ax + by, prove that \({{x}\over{a+x}}+{{y}\over{b+y}}+{{z}\over{c+z}}=1.\)

Determine the region in the plane determined by the inequalities.
\(2x+3y\le 6,\ x+4y\le 4,\ x\ge 0,\ y\ge 0.\)

The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day function for exchanging Singapore dollar to Indian Rupee is g(y) = 50.50y, Where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee

Find the domain of \(\frac { 1 }{ 1-2sinx } \)

Check whether the following for one-to-oneness and ontoness.
(i) \(f:R\rightarrow R\) defined by f(x) \(f(x)={1\over x}.\)
(ii) \(f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\) defined by \(f(x)=\frac{1}{x}\)

Compute log₃5 log₂₅27

For each given Angle, find a coterminal angle with a measure of \(\theta\) such that \(0^o\le \theta \le 360°\) 
395⁰ 

State whether the following sets are finite or infinite.
{x \(\in \) Z : x is even and less than 10}

Discuss the following relations for reflexivity, symmetricity and transitivity :
Let A be the set consisting of all the female members of a family. The relation R defined by "aRb if a is not a sister of b".

Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

Find the range of the following functions given by f(x) = 1 + 3 cos 2x.

Simplify \(\left( 125 \right) ^{ \frac { 2 }{ 3 } }\)

Which one of the following is not a singleton set?

The value of \({ log }_{ \sqrt { 2 } }512\) is

If ABCD is a cyclic quadrilateral then cos A + cos B + cos C + cos D = _______________

In ²ⁿC₃ : ⁿC₃ = 11 : 1 then n is

Expansion of \(log(\sqrt \frac{1+x}{1-x})\) is ______________

For any four sets A, B, C and D, which of the following is not true?

The number of roots of (x + 3)⁴+ (x + 5)⁴ = 16 is

If tan x = \(\frac { -1 }{ \sqrt { 5 } } \) and x lies in the IV quadrant, then the value of cos x is ___________

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

The coefficient of x⁵ in the series e^-2x is

The sequence \(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } }, \frac { 1 }{ \sqrt { 3 } +2\sqrt { 2 } },...... \)form an 

The sum up to n terms of the series \(\frac { 1 }{ \sqrt { 1 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 3 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 7 } } +\)....is 

The coefficient of x⁵ in the series e^-2x is

The term without x in \({ \left( 2x-\frac { 1 }{ 2{ x }^{ 2 } } \right) }^{ 12 }\) is ______________

The value of \({ 9 }^{ \frac { 1 }{ 3 } }\) ,\({ 9 }^{ \frac { 1 }{ 9 } }\)\({ 9 }^{ \frac { 1 }{ 27}}\),\(\infty \) is ______________

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

The number of five digit telephone numbers having at least one of their digits repeated is

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is _________

The product of r consecutive positive integers is divisible by _________

\(\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } } \)=

cos1⁰ + cos2⁰ + cos3⁰ +: : : + cos179⁰ =

\(\frac { sin(A-B) }{ cosAcosB } +\frac { sin(B-C) }{ cosBcosC } +\frac { sin(C-A) }{ cosCcosA } \) is

If tan A = \(\frac { a }{ a+1 } \) and B = \(\frac { 1 }{ 2a+1 } \) then the value of A + B is ___________

cos p = \(\frac { 1 }{ 7 } \) and cos Q = \(\frac { 13 }{ 14 } \) where P, Q are angles, then P-Q is _______________

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

The value of log_a b log_b c log_c a is

If 8 and 2 are the roots of x²+ ax + c = 0 and 3, 3 are the roots of x² + dx + b = 0; then the roots of the equation x²+ ax + b = 0 are

If |x + 3| ≥ 10 then ___________

The logarithmic form of 5² = 25 is ___________

The number of constant functions from a set containing m elements to a set containing n elements is

The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

Let A and B be subsets of the universal set N, the set of natural numbers. Then A^'∪[(A⋂B)∪B^'] is

If n(A) = 2 and n(B ∪ C) = 3, then n[(A \(\times\) B) ∪ (A \(\times\) C)] is

If two sets A and B have 17 elements in common, then the number of elements common to the set A \(\times\)B and B \(\times\)A is

The relation R defined on a set A= {0,-1, 1, 2} by xRy if |x²+y²| ≤ 2, then which one of the following is true?

Domain of the function \(y={x-1\over x+1}\) is __________

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

\(\sqrt [ 4 ]{ { \left( -2 \right) }^{ 4 } } \times { \left( -1000 \right) }^{ \frac { 1 }{ 3 } }\) is ___________

Which of the following is not true?

If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

Let A and B be subsets of the universal set N, the set of natural numbers. Then A^'∪[(A⋂B)∪B^'] is

If 3 is the logarithm of 343, then the base is

\(\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } } \)=

In \(\triangle\)ABC, \(\hat{C}\) = 90° then a cos A + b cos B is _______________

An integer is chosen at random from the first ten positive integers. Find the probability that it is (i) an even number (ii) multiple of three.

A die is rolled. If it shows an odd number, then find the probability of getting 5.

Suppose a fair die is rolled. Find the probability of getting (i) an even number (ii) multiple of three.

If A and B are two events associated with a random experiment for which P(A) = 0.35, P(A or B) = 0.85, and P(A and B) = 0.15. Find (i) P(only B) (ii) \(P(\bar{B})\) (iii) P(only A)

A die is thrown twice. Let A be the event, ‘First die shows 5’ and B be the event 'second die shows 5’. Find \(P(A\cup B)\) .

Evaluate \(\int { \frac { { x }^{ 4 }+{ x }^{ 2 }+1 }{ { x }^{ 2 }+x-1 } } \)dx

Evaluate \(\int { \frac { { sin }^{ 6 }x+cos^{ 6 }x }{ sin^{ 2 }xcos^{ 2 }x } } \)

Evaluate \(\int { \frac { \left( { a }^{ x }+{ b }^{ x } \right) ^{ 2 } }{ { a }^{ x }{ b }^{ x } } } \)dx

Evaluate if f'(x) = 3x² - \(\frac { 2 }{ { x }^{ 3 } } \) and f (1) = 0, find f (x)

Evaluate \(\int { \sqrt { 1+sinx } } \) dx, 0 < x < \(\frac { \pi }{ 2 } \)

Show that the function \(f\left( x \right) =\begin{cases} x-1,\quad x<2 \\ 2x-3,\quad x\ge 2 \end{cases}\)is not differentiable at x = 2.

Show  that\(f\left( x \right) ={ x }^{ 2 }\) is differentiable at x = 1 and find \(f^{ ' }\left( 1 \right) \)

Differentiate \(f\left( x \right) ={ e }^{ 2x }\)from first principles.

If \(y=\sqrt { x+1 } +\sqrt { x-1 } \) prove that\(\sqrt { { x }^{ 2 }+1 } \frac { dy }{ dx } =\frac { 1 }{ 2 } y.\)

If xy = 4, Prove that \(x\left( \frac { dy }{ dx } +{ y }^{ 2 } \right) =3y.\)

Calculate \(\lim _{ x\rightarrow0}{|x| } \).

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
\(lim_{x\rightarrow{1}}sin \pi x\)

The velocity in ft/sec of a falling object is modeled by \(r(t)=-\sqrt{32\over k}{1-e^{2t\sqrt{32k}}\over1+e^{-2r\sqrt{32k}}}\), where k is a constant that depends upon the size and shape of the object and the density of the air. Find the  limiting velocity of the object, that is, find \(lim_{t\rightarrow \infty}r(t).\)

Find the left and right limits of \(f(x)={x^2-4\over (x^2+4x+4)(x+3)}at \ x=-2\) .

Evaluate the following limits \(lim_{x\rightarrow\infty}{x^4-5x\over x^2-3x+1 }\)

Find the value of \(\lambda\) for which the vectors \(\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k} \) and \(\overrightarrow{b}=\overrightarrow{i}+\lambda \overrightarrow{j}+3\overrightarrow{k}\) are parallel.

Show that the following vectors are coplanar \(\hat{i}\) − 2\(\hat{j}\) + 3\(\hat{k}\), - 2\(\hat{i}\) + 3\(\hat{j}\) - 4\(\hat{k}\) ,-\(\hat{j}\) + 2\(\hat{k}\) .

Show that the following vectors are coplanar 5\(\hat{i}\) +6\(\hat{j}\) +7\(\hat{k}\) ,7 \(\hat{i}\) -8\(\hat{j}\) +9 \(\hat{k}\),3\(\hat{i}\)+20\(\hat{j}\) +5\(\hat{k}\) .

If \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\) prove that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular.

For any vector \(\overrightarrow{r}\) prove that \(\overrightarrow{r}\) = (\(\overrightarrow{r}.\hat{i}\)) \(\hat{i}\) + (\(\overrightarrow{r}.\hat{j}\)) \(\hat{j}\) + (\(\overrightarrow{r}.\hat{k}\)) \(\hat{k}\).

Prove that \(\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.\)

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of \(\begin{vmatrix} a & b & c \\ p & q & r \\ 1& 1 &1 \end{vmatrix}\)

Solve the following problems by using Factor Theorem :
Solve \(\begin{vmatrix} x+a &b &c \\ a & x+b & c \\ a & b &x+c \end{vmatrix}=0\)

Identify the singular and non-singular matrices:\(\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}\)

Identify the singular and non-singular matrices:\(\begin{bmatrix} 2&-3 &5 \\ 6 & 0 &4 \\ 1 & 5 & -7 \end{bmatrix}\)

Four persons are selected at random from a group of 3 men, 2 women, and 4 children. The probability that exactly two of them are children is

A man has 3 fifty rupee notes, 4 hundred rupees notes, and 6 five hundred rupees notes in his pocket. If 2 notes are taken at random, what are the odds in favour of both notes being of hundred rupee denomination?

A matrix is chosen at random from a set of all matrices of order 2, with elements 0 or 1 only. The probability that the determinant of the matrix chosen is non zero will be

If A and B are two events such that A ⊂ B and P(B)\(\neq o\), then which of the following is correct?

A number x is chosen at random from the first 100 natural numbers. Let A be the event of numbers which satisfies\({(x-10)(x-50)\over x-30}\ge0\), then P(A) is

If \(\int f(x) d x=g(x)+c\), then \(\int f(x) g^{\prime}(x) d x\)

\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\) is

\(\int \tan ^{-1} \sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}} d x\) is

\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x\) is

\(\int \frac{x^2+\cos ^2 x}{x^2+1} \operatorname{cosec}^2 x d x\) is

If the sum of the distance of a moving point in a plane from the axis is 1, then find the locus of the point.

Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, -1).

Find the equation of the straight line which passes through the intersection of the straight lines 2x + Y= 8 and 3x - 2y + 7 = 0 and is parallel to the straight line 4x+ y-11 =0.

Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of 120^o with the positive direction of x-axis.

Find the equation of the line which passes through the point (- 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5: 3 by this point.

Solve the quadratic equation 5^2x- 5^{x + 3}+ 125 = 5^x.

Solve \(\sqrt [ 8 ]{{{x}\over{x+3}} } -\sqrt{{{x+3}\over{x}}}=2.\)

A factory kept increasing its out-put by the same percentage every year. Find the percentage, if it is known that the output has doubled in the last two years.

Find the value of log₂ \(\left({{\sqrt [ 3 ]{4 } }\over{4^2\sqrt{8}}} \right).\)

Find x if \({{1}\over{2}}\) log₁₀ \((11+4\sqrt{7})\) = log₁₀ (2 + x).

Expand cos (A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = \(\frac{\pi}{2}\)

What must be the radius of a circular running path, around which an athlete must run 5 times in order to describe 1 km?

In a circular of diameter 40 cm, a chord is of length 20 cm. FInd the length of the minor arc of the chord?

If in two Circles, arcs of the same length subtend angles 60⁰ and 75⁰ at the center, find the ratio of their radii

Prove that sin 75^o - sin 15^o = cos 105^o + cos 15^o

Find the number of strings that can be made using all letters of the word THING. If these words are written as in a dictionary, what will be the 85^th string?

Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4 and 5 repetitions not allowed?

Find the sum of all 4-digit numbers that can be formed using digits 0, 2, 5, 7, 8 without repetition?

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three on the other side. Determine the number of ways in which the seating arrangement can be made?

If p(h) is the statement "n² + n is even" and if p(r) is true, then p(r + 1) is true.

Find \(\sqrt [ 3 ]{ 1001 } \) approximately. (two decimal places).

Prove that \(\sqrt [ 3 ]{ { x }^{ 3 }+6 } -\sqrt [ 3 ]{ { x }^{ 3 }+3 } \) is approximately equal to \(\frac { 1 }{ { x }^{ 2 } } \) when x is sufficiently large.

The first term of a G.P is 1. The sum of third and fifth terms is 90. Find the common ration of the G.P

Find all the sequence which are simultaneously arithmetic and geometric progression.

If the m^th term of a H.P is n and n^th term is m, then show that its p^th  term is \(\frac{mn}{p}\).

Graph the function f(x) = x³ and \(g(x)=\sqrt[3]x\) on the same co-ordinate plane. Find f o g and graph it on the plane as well. Explain your results.

Write the steps to obtain the graph of the function y = 3(x-1)²+5 from the graph y = x²

By taking suitable sets A, B, C, verify the following results:
(A\(\times\) B)\(\cap \)(B\(\times\)A) = (A\(\cap \)B) \(\times\) (B\(\cap \)A)

By taking suitable sets A, B, C, verify the following results:
C-(B-A) = (C\(\cap \) A) \(\cup \) (C\(\cap \)B')

If A\(\times\) A has 16 elements, S = {(a, b) \(\in \) A\(\times\) A:a < b}; (−1, 2) and (0, 1) are two elements of S, then find the remaining elements of S.

If \(y={1\over a-z}\) , then \({dz\over dy}\) is

If y = mx + c and f(0) =\(f '(0)=1\), then f(2) is

\({d\over dx}(e^{x+5log \ x})\) is

\(x={1-t^2\over 1+t^2},y={2t\over 1+t^2}\) then \({dy\over dx}\)is

The differential coefficient of log₁₀ x with respect to log_x10 is

\(lim_{x\rightarrow\infty}{sin \ x \over x} \)

If f(x) = x(-1)^{\(\left\lfloor 1\over x \right\rfloor \)}, \(x\le0\), then the value of \(lim_{x\rightarrow 0}f(x)\) is equal to

If \(lim_{x \rightarrow 0}{sin \ px\over tan \ 3x}=4\) , then the value of p is

\(lim_{x \rightarrow 0}{e^{tan \ x}-e^x\over tan x-x}=\)

The function \(f(x)= \begin{cases}\frac{x^{2}-1}{x^{3}+1} & x \neq-1 \\ P & x=-1\end{cases}\)is not defined for x = −1. The value of f(−1) so that the function extended by this value is continuous is

If A = \(\begin{bmatrix}\lambda & 1 \\ -1 & -\lambda \end{bmatrix}\), then for what value of \(\lambda\), A² = O?

If A is a square matrix, then which of the following is not symmetric?

If A and B are symmetric matrices of order n, where (A \(\neq\) B), then

If the points (x,−2), (5, 2), (8, 8) are collinear, then x is equal to

If A is skew-symmetric of order n and C is a column matrix of order n \(\times\) 1, then C^T AC is

The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between \(\overrightarrow{OP}\)and the z-axis is

One of the diagonals of parallelogram ABCD with \(\overrightarrow{a}\) and \(\overrightarrow{b}\) as adjacent sides is \(\overrightarrow{a}+\overrightarrow{b}\) The other diagonal \(\overrightarrow{BD}\) is

The value of  \(\theta \in (0,{\pi\over 2})\) for which the vectors \(\overrightarrow{a}=(sin \theta)\hat{i}+(cos\theta)\hat{j}\) and \(\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}\) are perpendicular, is equal to

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible
\(P(A)=\frac { 2 }{ 5 } ,\quad P(B)=\frac { 1 }{ 5 } ,\quad P(C)=\frac { 3 }{ 5 } \)

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible.
P(A) = 0.421, P(B) = 0.527  P(C) = 0.042

There are two identical urns containing respectively 6 black and 4 red balls, 2 black and 2 red balls. An urn is chosen at random and a ball is drawn from it.
(i) find the probability that the ball is black
(ii) if the ball is black, what is the probability that it is from the first urn?

If two coins are tossed simultaneously, then find the probability of getting (i) one head and one tail (ii) at most two tails

Five mangoes and 4 apples are in a box. If two fruits are chosen at random, find the probability that (i) one is a mango and the other is an apple (ii) both are of the same variety.

Integrate the following with respect to x : \(\frac{1}{x^{10}}\)

Integrate the following with respect to x : \(\sqrt{x}\)

IIntegrate the following with respect to x : \({cot \ x \over sin \ x}\)

Integrate the following with respect to x :\({1\over x^3}\)

Evaluate the following with respect to x : \(\int{\sqrt{(15-2x)}}dx\)

Differentiate the following with respect to x : \(y=(x-{1\over x})^2\)

Differentiate the following with respect to x : y = xe^x log x

Find the derivatives of the following functions with respect to corresponding independent variables: y = sin x + cos x

Differentiate the following: y = cos (tan x)

Differentiate the following: \(f(t)=\sqrt[3]{1+\tan t}\)

In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow{0}}{\sqrt{x+3}-\sqrt{3}\over x}\)

In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow 0}{sin x\over x}\)

Compute \(lim_{x\rightarrow8}(5x)\)

Compute\(lim_{x\rightarrow-2}(-{3\over 2}x)\)

Find the positive integer n so that \(lim_{x\rightarrow 3}{x^n-3^n\over x-3}=27\)