If P(A) = \(\frac12\), P(B) = 0, then P(A|B) is ______.

Let f : R ➝ R be defined as f (x) = 3x. Choose the correct answer

Consider a binary operation * on N defined as a * b = a³ + b³. Choose the correct answer

If A = {1,3,5,7} and define a relation, such that R = { (a,b) a,b ∈ A : |a+b| = 8}. Then how many elements are there in the relation R

 If A = diag(3, -1), then matrix A is

The number of all possible matrices of order 3 × 3 with each entry

If A, B are symmetric matrices of same order, then AB – BA is a

For what real value of y will matrix A be equal to matrix B, where
\(A=\begin{bmatrix} 3x-4 & 5y \\ 8 & { y }^{ 2 }-4y \end{bmatrix};B=\begin{bmatrix} x+1 & 6{ y }^{ 2 }+1 \\ 8 & -3 \end{bmatrix}\)

A function \(f(x)=\begin{cases} \frac { sinx }{ x } +cosx,x\neq 0 \\ 2k\quad \quad \quad \quad ,x=0 \end{cases}\) is continuous at x = 0 for

If y = tan^-1 \(\left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) \), then \(\frac { dy }{ dx } \) is equal to

The equation of the normal to the curve y = sin x at (0, 0) is

The absolute maximum value of y = x³ – 3x + 2 in 0 ≤ x ≤ 2 is

The interval in which y = x² e^–x is increasing is

For all real values of x, the minimum value of \(\frac { 1-x+{ x }^{ 2 } }{ 1+x+{ x }^{ 2 } } \) is

Let \({ I }_{ 1 }=\int _{ 1 }^{ 2 }{ \frac { dx }{ \sqrt { 1+{ x }^{ 2 } } } } \) and \({ I }_{ 2 }=\int _{ 1 }^{ 2 }{ \frac { dx }{ x } } \), then

\(\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\) is equal to 

Area of a rectangle having vertices A, B, C and D with position vectors \(-\widehat { i } +\frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \), \(\widehat { i } +\frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \), \(\widehat { i } -\frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \) and \(-\widehat { i } -\frac { 1 }{ 2 } \widehat { j } +4\widehat { k } \), respectively is

If a, b, c and d are the position vectors of the points A, B, C and D such that a + c = b + d, then ABCD is a

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is _____.

Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is

If f(x) = 27x³ and g(x) = x^1/3 find gof(x).

If \(2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix}+\begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix}=\begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}\), then write the value of (x+y).

If the following matrix is skew symmetric, find the values of a, b, c
\(A=\left[ \begin{matrix} 0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0 \end{matrix} \right] \)

Show that the function f (x) = \(\begin{cases} { x }^{ 3 }+3\quad ,\quad if\quad x\neq 0 \\ 1\quad \quad \quad ,\quad if\quad x=0 \end{cases}\) is not continuous at x = 0. 

The total cost C(x) in Rupees, associated with the production of x units of an item is given by
C(x) = 0.005 x³ – 0.02 x² + 30x + 5000
Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Evaluate the integral: \(({\int{\sqrt x+{1\over\sqrt x}}})^2dx.\)

\(\int {1+tan\ x\over 1-tan\ x}dx\).

\(\int tan^{-1}(cot\ x)dx.\)

What is the cosine of the angle which the vector \(\sqrt { 2 } \hat { i } +\hat { j } +\hat { k } \) makes with y-axis?

Find the scalar components of the vector \(\overset\rightarrow {AB} \) with initial point A(2, 1) and terminal point B(-5, 7).

Find \(\lambda\) and \(\mu\) if \((\overset\wedge i+3\overset\wedge j+9\overset\wedge k)\times(3\overset\wedge i-\lambda \overset\wedge j+\mu \overset\wedge k)=0\)

Given P(A) = 0.4, P(B) = 0.7 and P(B/A) = 0.6, Find \(P(A\cup B)\)

Let f and g be real function be \(f(x)=\sqrt { x+4 } ,x\ge 4\) find the function fg, \(\frac { f }{ g } \)

Find the value of X and Y if
\(X+Y=\left[ \begin{matrix} 2 & 3 \\ 5 & 1 \end{matrix} \right] ,X-Y=\left[ \begin{matrix} 6 & 5 \\ 7 & 3 \end{matrix} \right] \)

Find the value of x and y in each if AB exist
(i) \({ A }_{ 3\times x },{ B }_{ 4\times y }\) and \({ AB }_{ 3\times 3 }\)
(ii) \({ A }_{ x\times 2 },{ B }_{ y\times 4 }\) and \({ AB }_{ 3\times 4 }\)

If y = tan^-1\(\sqrt { \frac { 1-x }{ 1+x } } find\frac { dy }{ dx } \)

If x = \(\frac { at }{ 1+{ t }^{ 2 } } ,y=\frac { a{ t }^{ 2 } }{ 1+{ t }^{ 2 } } ,find\frac { dy }{ dx } at\) t = 2

Find the point on the curve \(y^2 =8x + 3\)  for which the y-coordinate changes 4 times more than coordinate of x.

\(\int { { sin }^{ 2 }x{ \ cos }^{ 2 }x } dx\)

Find the unit vector in the direction of \(\overset\rightarrow a+\overset\rightarrow b\)if \(\overset\rightarrow a= 2\overset\wedge i+\overset\wedge j+3\overset\wedge k\), and \(\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k\)

If P(E) = \(\frac { 7 }{ 13 } \) , P(F) = \(\frac { 9 }{ 13 } \) and P(E\(\cap\)F) = \(\frac { 4 }{ 13 } \),then evaluate :
(a) \(P(\overline { E } /F)\)  
(b) \(P(\overline { E } /F)\)

Show that the relation R in the set Z of integers given by:
R = {(a, b) : 2 divides a-b} is an equivalence relation

(Diet Problem) A dietician has to develop a special diet using two foods P and Q. Each packet (containing 50g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet. What is the minimum amount of vitamin A?

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) \(f:R\rightarrow R\) defined by f(x) = 3 - 4x
(ii) \(f:R\rightarrow R\) defined by \(f(x)=1+x^{ 2 }\) .

Find X, if \(Y=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\) and \(2X+Y=\begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}\) .

Discuss the continuity of the function f given by:
\(f(x)=\left| x \right| at\quad x=0.\)

Discuss the continuity of the function f defined by:
\(f(x)={ x }^{ 3 }+{ x }^{ 2 }-1\)

Differentiate \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -1 }{ x } \right) \) w.r.t.x.

Prove that the function given by by f (x) = cos x is
(a) strictly decreasing in \((0,\pi )\)
(b) strictly increasing in \((\pi ,2\pi )\)
(c) neither increasing nor decreasing in \((0,2\pi )\)

Find the absolute maximum and absolute value of the function given by:
\(f(x)sin^{ 2 }x-cosx,x\in [0,\pi ]\)

Let \(\vec{a}=\hat{i}+2 \hat{j} \text { and } \vec{b}=2 \hat{i}+\hat{j} . \text { Is }|\vec{a}|=|\vec{b}| ?\)  Are the vector \(\overset { \rightarrow }{ a } \) is \(\overset { \rightarrow }{ b } \) equal?

Six balls are drawn successively from an urn containing 7 red and 9 black balls.Tell whether or not the trials of drawing black balls are Bernoulli trials when after each draws the ball drawn is:
(i) replaced
(ii) not replaced in the urn.

Two numbers are selected at random (without replacement)from first six positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution

Show that the relation S in the set R of real numbers, defined as
S = {(a, b): a,b \(\in\)R and a \(\le\)b³} is neither reflexive, nor symmetric nor transitive.

Check the injectivity and surjectivity of the following
(i) f : N \(\rightarrow\) N given by f(x) = x²
(ii) f : R \(\rightarrow\) R given by f(x) = x²
(iii) f : Z → Z given by f(x) = x²
(iv) f : N → N given by f(x) = x³
(v) f : Z → Z given by f(x) = x³

Using elementary transformations, find the inverse of the matrix

\(\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{matrix} \right] \).

Find the value of a for which the function f defined by

\(f(x)=\begin{cases} a\quad sin\frac { \pi }{ 2 } (x+1)\quad \quad ,\quad x\le 0 \\ \frac { tan\quad x\quad -\quad sin\quad x }{ { x }^{ 3 } } \ \ \ \ ,\quad x>0 \end{cases}\) is continuous at x = 0. 

Let f be a function defined on an open interval I.(First Derivative Test)