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

(a)

one-to-one

(b)

on to

(c)

bijection

(d)

cannot be defined

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

(a)

[-9,9]

(b)

R

(c)

[-3,3]

(d)

[0,9]

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

(a)

\({ f }^{ -1 }(x)=\begin{cases} x\quad if\quad x<1 \\ \sqrt { x } \quad if\quad 1\le x\le 16 \\ \frac { { x }^{ 2 } }{ 64 } \quad if\quad x>16 \end{cases}\)

(b)

\({ f }^{ -1 }(x)=\begin{cases} -x\quad if\quad x<1 \\ \sqrt { x } \quad if\quad 1\le x\le 16 \\ \frac { { x }^{ 2 } }{ 64 } \quad if\quad x>16 \end{cases}\)

(c)

\({ f }^{ -1 }(x)=\begin{cases} { x }^{ 2 }\quad if\quad x<1 \\ \sqrt { x } \quad if\quad 1\le x\le 16 \\ \frac { { x }^{ 2 } }{ 64 } \quad if\quad x>16 \end{cases}\)

(d)

\({ f }^{ -1 }(x)=\begin{cases} { 2x }\quad if\quad x<1 \\ \sqrt { x } \quad if\quad 1\le x\le 16 \\ \frac { { x }^{ 2 } }{ 8 } \quad if\quad x>16 \end{cases}\)

Let f:R➝R be defined by f(x) = 1 - |x|. Then the range of f is

(a)

R

(b)

(1,∞)

(c)

(-1,∞)

(d)

(-∞,1]

If |x+2| \(\le\) 9, then x belongs to

(a)

\((-\infty ,-7)\)

(b)

[-11, 7]

(c)

\((-\infty ,-7)\cup (11,\infty)\)

(d)

(-11, 7)

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

(a)

xb < yb

(b)

xb > yb

(c)

xb ≤ yb

(d)

\(\frac { x }{ b } \ge \frac { y }{ b } \)

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

(a)

1

(b)

2

(c)

3

(d)

4

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

(a)

\(\frac { b }{ a } \)

(b)

\(\frac { a }{ b } \)

(c)

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

(d)

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

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 

(a)

\(\sqrt { 2n+1 } \)

(b)

\(\frac { \sqrt { 2n+1 } }{ 2 } \)

(c)

\(\sqrt { 2n+1 } -1\)

(d)

\(\frac { \sqrt { 2n+1 } -1 }{ 2 } \)

\(\sqrt \frac{1-2x}{1+2x}\) is approximately equal to ______________

(a)

1- 2x-x²

(b)

1 + 2x+ x²

(c)

1+ 2x

(d)

1-2x+x²

If a vertex of a square is at the origin and its one side lies along the line 4x + 3y - 20 = 0, then the area of the square is

(a)

20 sq. units

(b)

16 sq. units

(c)

25 sq. units

(d)

4 sq.units

The length of the perpendicular from origin to line is \(\sqrt{3}x-y+24=0\) is ______________

(a)

2\(\sqrt{3}\)

(b)

8

(c)

24

(d)

12

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

(a)

(2, - 1)

(b)

(- 2, 1)

(c)

(2, 1)

(d)

(- 2, - 1)

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂, y₂), (x₃, y₃ ) are

(a)

vertices of an equilateral triangle

(b)

vertices of a right angled triangle

(c)

vertices of a right angled isosceles triangle

(d)

collinear

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

(a)

\(cos^{-1}({1\over 3})\)

(b)

\(cos^{-1}({2\over 3})\)

(c)

\(cos^{-1}({1\over\sqrt 3})\)

(d)

\(cos^{-1}({2\over\sqrt 3})\)

If the projection of \(5\hat{i}-\hat{j}-3\hat{k}\) on the vector \(\hat{i}+3\hat{j}+\lambda\hat{k}\) is same as the projection of \(\hat{i}+3\hat{j}+\lambda\hat{k}\) on \(5\hat{i}-\hat{j}-3\hat{k}\), then \(\lambda\) is equal to

(a)

\(\pm 4\)

(b)

\(\pm 3\)

(c)

\(\pm 5\)

(d)

\(\pm 1\)

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

(a)

2

(b)

1

(c)

-2

(d)

0

If f(x) = x² - 3x, then the points at which f(x) = f '(x) are

(a)

both positive integers

(b)

both negative integers

(c)

both irrational

(d)

one rational and another irrational

If \(f(x)= \begin{cases}2 a-x, & \text { for } \quad-a<x<a \\ 3 x-2 a & \text { for } \quad x \geq a\end{cases}\), then which one of the following is true?

(a)

f(x) is not differentiable at x = a

(b)

f(x) is discontinuous at x = a

(c)

f(x) is continuous for all x in R

(d)

f(x) is differentiable for all x \(\ge\) a

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

(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)

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

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

(a)

e^x tan^-1(x+1)+c

(b)

tan^-1(e^x)+c

(c)

\(e^x{(tan^{-1}x)^2\over 2}+c\)

(d)

e^x tan^-1 x+c

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)

\({19\over 90}\)

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

(a)

\({3\over 16}\)

(b)

\({3\over 8}\)

(c)

\({1\over 4}\)

(d)

\({5\over 8}\)

The probability of two events A and B are 0.3 and 0.6 respectively. The probability that both A and B occur simultaneously is 0.18. The probability that neither A nor B occurs is

(a)

0.1

(b)

0.72

(c)

0.42

(d)

0.28

If m is a number such that m \(\le\) 5, then the probability that quadratic equation 2x² + 2mx + m + 1 = 0 has real roots is

(a)

\({1\over 5}\)

(b)

\({2\over 5}\)

(c)

\({3\over 5}\)

(d)

\({4\over 5}\)