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

(a)

A and B are two matrices not necessarily of same order

(b)

A and B are square matrices of same order

(c)

Number of columns of A is equal to the number of rows of B

(d)

A = B.

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.

(a)

1 + \(\alpha ^2+\beta \gamma=0\)

(b)

1 - \(\alpha ^2-\beta \gamma=0\)

(c)

1 - \(\alpha ^2+\beta \gamma=0\)

(d)

1 + \(\alpha ^2-\beta \gamma=0\)

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

(a)

\(\begin{bmatrix} -5& -4 \\ 8 & -9 \end{bmatrix}\)

(b)

\(\begin{bmatrix} -5& 4 \\ -8 & 9 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 5& 4 \\ 8 & 9 \end{bmatrix}\)

(d)

\(\begin{bmatrix} -5& -4 \\ -8 & -9 \end{bmatrix}\)

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

(a)

27

(b)

81

(c)

135

(d)

405

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

(a)

\(\overrightarrow{a}-\overrightarrow{b}\)

(b)

\(\overrightarrow{b}-\overrightarrow{a}\)

(c)

\(\overrightarrow{a}+\overrightarrow{b}\)

(d)

\(\overrightarrow{a}+\overrightarrow{b}\over 2\)

Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to

(a)

225

(b)

275

(c)

325

(d)

300

\(lim_{x \rightarrow 0}{a^x-b^x\over x}=\)

(a)

log ab

(b)

log\(({a\over b})\)

(c)

log\(({b\over a})\)

(d)

\({a\over b}\)

The value of \(lim_{x\rightarrow k^-}x-\left\lfloor x \right\rfloor \) , where k is an integer is

(a)

-1

(b)

1

(c)

0

(d)

2

\(\lim _{ x\rightarrow 2 }{ \frac { 2{ x }^{ 2 }+x+1 }{ x+2 } } \) is equal to

(a)

\(\frac { 1 }{ 2 } \)

(b)

2

(c)

\(\frac { 11 }{ 4 } \)

(d)

0

The rate of change of area A of a circle of radius r is

(a)

\(2\pi r\)

(b)

\(2\pi r\frac { dr }{ dt } \)

(c)

\(\pi { r }^{ 2 }\frac { dr }{ dt } \)

(d)

\(\pi \frac { dr }{ dt } \)

Find the odd one of out of the following

(a)

tan x

(b)

\(\frac{1}{x}\)

(c)

\(\\ \\ \\ \\ \\ \\ \\ \frac { { x }^{ 2 }+5x+4 }{ { x }^{ 2 }+4x+4 } \)

(d)

cos x

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

(a)

1

(b)

2

(c)

3

(d)

-3

\(\text { If } f(x)=\left\{\begin{array}{ll} a x^2-b, & -1<x<1 \\ \frac{1}{|x|}, & \text { elsewhere } \end{array} \ \text { is differentiable at } x=1\right. \text {, then }\) 

(a)

\(a={1\over2},b={-3\over 2}\)

(b)

\(a={-1\over2},b={3\over 2}\)

(c)

\(a=-{1\over2},b=-{3\over 2}\)

(d)

\(a={1\over2},b={3\over 2}\)

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\log \left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{dy}{dx}\) is ______

(a)

\(\frac { 4{ x }^{ 3 } }{ 1-{ x }^{ 4 } } \)

(b)

\(-\frac { 4x }{ 1-{ x }^{ 4 } } \)

(c)

\(\frac { 1 }{ 4-{ x }^{ 4 } } \)

(d)

\(\frac { -4{ x }^{ 3 } }{ 1-{ x }^{ 4 } } \)

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) then \(\frac{dy}{dx}\) is ______

(a)

\(\frac { -2 }{ 1+{ x }^{ 2 } } \)

(b)

\(\frac { 2 }{ 1+{ x }^{ 2 } } \)

(c)

\(\frac { 1 }{ 2-{ x }^{ 2 } } \)

(d)

\(\frac { 2 }{ 2-{ x }^{ 2 } } \)

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

(a)

\({1\over2}sin 2x+c\)

(b)

\(-{1\over2}sin 2x+c\)

(c)

\({1\over2}cos 2x+c\)

(d)

\(-{1\over2}cos 2x+c\)

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

(a)

\(\sqrt{x^2-1}-2 log|x+\sqrt{x^2-1}|+c\)

(b)

\(sin^{-1}x-2 log|x+\sqrt{x^2-1}|+c\)

(c)

\(2 log|x+\sqrt{x^2-1}|-sin^{-1}x+c\)

(d)

\(\sqrt{x^2-1}+2log|x+\sqrt{x^2-1}|+c\)

\(\int { \frac { 1 }{ 2 } } { sec }^{ 2 }\) x dx is _____ + c.

(a)

\(\frac { 1 }{ 2 } \) tan x

(b)

tan x

(c)

2 tan x

(d)

none of these

\(\int { \frac { 4\left( sin^{ -1 }x \right) ^{ 3 } }{ \sqrt { 1-{ x }^{ 2 } } } } \) dx = ________+c.

(a)

log (sin ^-1x)

(b)

(sin ^-1 x)⁴

(c)

4 (sin^-1 x)⁴

(d)

\(\frac { \left( { sin }^{ -1 }x \right) ^{ 4 } }{ 4 } \)

\(\int { \frac { 1 }{ 9x^{ 2 }-4 } } \) dx = ________+c.

(a)

log\(\left| \frac { 3x-2 }{ 3x+2 } \right| \)

(b)

\(\frac { 1 }{ 12 } \)log\(\left| \frac { 3x-2 }{ 3x+2 } \right| \)

(c)

12log\(\left| \frac { 3x-2 }{ 3x+2 } \right| \)

(d)

\(\frac { 1 }{ 12 } log\left| \frac { 3x+2 }{ 3x-2 } \right| \)

A letter is taken at random from the letters of the word ‘ASSISTANT’ and another letter is taken at random from the letters of the word ‘STATISTICS’. The probability that the selected letters are the same is

(a)

\({7\over 45}\)

(b)

\({17\over 90}\)

(c)

\({29\over 90}\)

(d)

\({19\over 90}\)

Three integers are chosen at random from the first 20 integers. The probability that their product is even is

(a)

\(\frac { 2 }{ 19 } \)

(b)

\(\frac { 3 }{ 19 } \)

(c)

\(\frac { 17 }{ 19 } \)

(d)

\(\frac { 4 }{ 19 } \)

If P(A\(\cup \)B) = 0.8 and P(A\(\cap \)B) = 0.3 then \(P(\bar { A } )+P(\bar { B } )\) =

(a)

0.3

(b)

0.5

(c)

0.7

(d)

0.9

Two dice are thrown. It is known that the sum of the numbers on the dice was less than 6, the probability of getting a sum 3 is

(a)

\(\frac { 1 }{ 18 } \)

(b)

\(\frac { 5 }{ 18 } \)

(c)

\(\frac { 1 }{ 5 } \)

(d)

\(\frac { 2 }{ 5 } \)

Assertion (A) : nP_r > nC_r
Reason (R) : nP_r = nC_r x r!

(a)

Both A and (R) are true and (R) is the correct explanation of (A)

(b)

Both A and R are true but (R) is not the correct explanation of A

(c)

A is true R is false

(d)

A is false R is true