For the set A = {1, 2, 3}, define a relation R in the set A as follows
R = {(1, 1), (2,2), (3, 3), (1, 3)}
Then, the ordered pair to be added to R to make it the smallest equivalence relation is

1

(a)

(1, 3)

(b)

(3, 1)

(c)

(2, 1)

(d)

(1, 2)

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

1

(a)

720

(b)

120

(c)

0

(d)

None of these

The matrix \(\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]\) is a

1

(a)

diagonal matrix

(b)

symmetric matrix

(c)

skew-symmetric matrix

(d)

scalar matrix

On using elementary column operations \(C_{2} \rightarrow C_{2}-2 C_{1}\) in the following matrix equation \(\left[\begin{array}{cc}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]\), we have

1

(a)

\(\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]\)

(b)

\(\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ -0 & 2\end{array}\right]\)

(c)

\(\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -3 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -2 & 4\end{array}\right]\)

(d)

\(\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]\)

If matrix \(A=\left[a_{i j}\right]_{2 \times 2},\ where \ a_{i j}=\left\{\begin{array}{l}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.\) Then \(A^{2}\) is equal to 

1

(a)

I

(b)

A

(c)

0

(d)

None ofthese

For any two matrices A and B, we have

1

(a)

AB=BA

(b)

AB ≠BA

(c)

AB = 0

(d)

None of these

If \(A=\left|\begin{array}{llr} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right|\) then A^-I exists, if

1

(a)

\(\lambda=2\)

(b)

\(\lambda \neq 2\)

(c)

\(\lambda \neq-2\)

(d)

None of these

If \(f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\) is equal to

1

(a)

0

(b)

-1

(c)

2

(d)

3

A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/s, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is

1

(a)

\(\frac{1}{10} \mathrm{rad} / \mathrm{s}\)

(b)

\(\frac{1}{20} \mathrm{rad} / \mathrm{s}\)

(c)

\(20 \mathrm{rad} / \mathrm{s}\)

(d)

10 rad/s

The interval on which the function \(f(x)=2 x^{3}+9 x^{2}+12 x-1\) is decreasing is

1

(a)

\([-1, \infty)\)

(b)

\([-2,-1]\)

(c)

\((-\infty,-2]\)

(d)

[-1,1]

If \(y=x(x-3)^{2}\) decreases for the values of x given by 

1

(a)

\(1<x<3\)

(b)

\(x<0\)

(c)

\(x>0\)

(d)

\(0<x<\frac{3}{2}\)

The function \(f(x)=\tan x-x\) 

1

(a)

always increases

(b)

always decreases

(c)

never increases

(d)

sometimes increases and sometimes decreases

Which of the following functions is decreasing on \(\left(0, \frac{\pi}{2}\right)\)?

1

(a)

sin2x

(b)

tanx

(c)

cosx

(d)

cos3x

The curve  \(y=x^{1 / 5}\) has at (0, 0)

1

(a)

a vertical tangent (parallel to Y-axis)

(b)

a horizontal tangent (parallel to X-axis)

(c)

an oblique tangent

(d)

no tangent

The tangent to the curve \(y=e^{2 x}\) at the point (0,1) meets X-axis at

1

(a)

(0,1

(b)

\(\left(-\frac{1}{2}, 0\right)\)

(c)

(2,0)

(d)

(0,2)

\(\int \frac{x+\sin x}{1+\cos x} d x\) is equal to

1

(a)

\(\log |1+\cos x|+C\)

(b)

\(\log |x+\sin x|+C\)

(c)

\(x-\tan \frac{x}{2}+C\)

(d)

\(x \cdot \tan \frac{x}{2}+C\)

\(\int_{0}^{\pi / 2} \cos x e^{\sin x} d x\) is equal to 

1

(a)

e + 1

(b)

e -1

(c)

e

(d)

-e

\(\int_{a+c}^{b+c} f(x) d x\) is equal to

1

(a)

\(\int_{a}^{b} f(x-c) d x\)

(b)

\(\int_{a}^{b} f(x+c) d x\)

(c)

\(\int_{a}^{b} f(x) d x\)

(d)

\(\int_{a-c}^{b-c} f(x) d x\)

The magnitude of the vector \(6 \hat{i}+2 \hat{j}+3 \hat{k}\) is

1

(a)

5

(b)

7

(c)

12

(d)

1

The position vector of the point which divides the join of points with position vectors \(\vec{a}+\vec{b}\) and \(2 \vec{a}-\vec{b}\) in the ratio 1: 2 is

1

(a)

\(\frac{3 \vec{a}+2 \vec{b}}{3}\)

(b)

\(\vec{a}\)

(c)

\(\frac{5 \vec{a}-\vec{b}}{3}\)

(d)

\(\frac{4 \vec{a}+\vec{b}}{3}\)

The angle between two vectors \(\vec{a} \text { and } \vec{b}\) with magnitude, \(\sqrt{3} \text { and } 4\) respectively and \(\vec{a} \cdot \vec{b}=2 \sqrt{3}\) is 

1

(a)

\(\frac{\pi}{6}\)

(b)

\(\frac{\pi}{3}\)

(c)

\(\frac{\pi}{2}\)

(d)

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

If \(|\vec{a}|=10,|\vec{b}|=2\) and \(\vec{a} \cdot \vec{b}=12\) 12, then the value of \(|\vec{a} \times \vec{b}|\) is 

1

(a)

5

(b)

10

(c)

14

(d)

16

The coordinates of the foot of the perpendicular drawn from the point (2. 5. 7) on the X-axis are given by 

1

(a)

(2.0.0)

(b)

(0,5,0)

(c)

(0,0,7)

(d)

(0,5,7)