If a relation R on the set {1,2, 3} be defined by R = {(1, 2)}, then R is

1

(a)

reflexive

(b)

transitive

(c)

symmetric

(d)

None of these

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

 Let \(f: R \rightarrow R\) be the functions defined by \(f(x)=x^{3}+5\). Then, \(f^{-1}(x)\) is 

1

(a)

\((x+5)^{1 / 3}\)

(b)

\((x-5)^{1 / 3}\)

(c)

\((5-x)^{1 / 3}\)

(d)

5-x

If A and B are two matrices of the order \(3 \times m\) and \(3 \times n\) respectively and m=n, then the order of the matrix \((5 A-2 B)\) is 

1

(a)

m x 3

(b)

3 x 3

(c)

m x n

(d)

3 x n

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

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

Let \(\Delta=\left|\begin{array}{lll} A x & x^{2} & 1 \\ B y & y^{2} & 1 \\ C z & z^{2} & 1 \end{array}\right| \text { and } \Delta_{1}=\left|\begin{array}{ccc} A & B & C \\ x & y & z \\ z y & z x & x y \end{array}\right|\) then

1

(a)

\(\Delta_{1}=-\Delta\)

(b)

\(\Delta \neq \Delta_{1}\)

(c)

\(\Delta^{2}-\Delta_{1}=0\)

(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

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 function \(f(x)=\tan x-x\) 

1

(a)

always increases

(b)

always decreases

(c)

never increases

(d)

sometimes increases and sometimes decreases

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

1

(a)

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

(b)

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

(c)

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

(d)

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

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 vector having initial and terminal points as \((2,5,0) \text { and }(-3,7,4)\) respectively is

1

(a)

\(-\hat{i}+12 \hat{j}+4 \hat{k}\)

(b)

\(5 \hat{i}+2 \hat{j}-4 \hat{k}\)

(c)

\(-5 \hat{i}+2 \hat{j}+4 \hat{k}\)

(d)

\(\hat{i}+\hat{j}+\hat{k}\)

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)

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

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)

If the plane \(2 x-3 y+6 z-11=0\) makes an angle \(\sin ^{-1} \alpha\) with X-axis, then the value of \(\alpha\) is

1

(a)

\(\frac{\sqrt{3}}{2}\)

(b)

\(\frac{\sqrt{2}}{3}\)

(c)

\(\frac{2}{7}\)

(d)

\(\frac{3}{7}\)

The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15,15), (0, 20). Let Z = px + qy where \(q>0\) .Then, the condition on p and q, so that the maximum of Z occurs at both the
points (15,15) and (0, 20) is

1

(a)

p = q

(b)

p = 2q

(c)

q = 2p

(d)

q = 3p

If 3 A and B are two events such that \(P(B)=\frac{3}{5}\),\(P(A / B)=\frac{1}{2} \text { and } P(A \cup B)=\frac{4}{5}\) then P(A) equals

1

(a)

\(\frac{3}{10}\)

(b)

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

(c)

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

(d)

\(\frac{3}{5}\)

If a die is thrown and a card is selected at random from a deck of 52 playing cards, then the probability of getting an even number on the die and a spade card is

1

(a)

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

(b)

\(\frac{1}{4}\)

(c)

\(\frac{1}{8}\)

(d)

\(\frac{3}{4}\)