If f(x) = |x - 2| + |x + 2|, x ∈ R, then

(a)

\(f(x)=\left\{\begin{array}{lll} -2 x & \text { if } & x \in(-\infty,-2] \\ 4 & \text { if } & x \in(-2,2] \\ 2 x & \text { if } & x \in(2, \infty) \end{array}\right.\)

(b)

\(f(x)=\begin{cases}2x\ if\ x∈(-∞,-2] \\4x\ if \ x∈(-2,2]\\ - 2x\ if\ x∈(2,∞)\end{cases}\)

(c)

\(f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\-4x\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}\)

(d)

\(f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\2x\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}\)

Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

(a)

T is an equivalence relation but S is not an equivalence relation

(b)

Neither S nor T is an equivalence relation

(c)

Both S and T are equivalence relation

(d)

S is an equivalence relation but T is not an equivalence relation.

The solution set of the following inequality |x-1| \(\ge\) |x-3| is

(a)

[0, 2]

(b)

\([2,\infty)\)

(c)

(0, 2)

(d)

\((-\infty,2)\)

The value of \({ log }_{ \sqrt { 2 } }512\) is

(a)

16

(b)

18

(c)

9

(d)

12

If \(\pi <2\theta <\frac { 3\pi }{ 2 } \), then \(\sqrt { 2+\sqrt { 2+2cos4\theta } } \) equals to

(a)

-2 cosፀ

(b)

-2 sinፀ

(c)

2 cosፀ

(d)

2 sinፀ

If sin \(\theta\) + cos \(\theta\) = 1 then sin⁶ \(\theta\) + cos⁶ \(\theta\) is _______________

(a)

1

(b)

0

(c)

-1

(d)

2

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

(a)

125

(b)

124

(c)

64

(d)

63

In ²ⁿC₃ : ⁿC₃ = 11 : 1 then n is

(a)

5

(b)

6

(c)

11

(d)

7

^(n-1)C_r + ^(n-1)C_(r-1) is

(a)

⁽ⁿ⁺¹⁾C_r

(b)

^(n-1)C_r

(c)

ⁿC_r

(d)

ⁿC_r-1

The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is

(a)

⁵²C₅

(b)

⁴⁸C₅

(c)

⁵²C₅ + ⁴⁸C₅

(d)

⁵²C₅ - ⁴⁸C₅

1+3+5+7+........+17 is equal to

(a)

101

(b)

81

(c)

71

(d)

61

If a, 8, b are in AP, a, 4, b are in GP, and if a, x, b are in HP then x is

(a)

2

(b)

1

(c)

4

(d)

16

The coefficient of x⁵ in the series e^-2x is

(a)

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

(b)

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

(c)

\(\frac { -4 }{ 15 } \)

(d)

\(\frac { 4 }{ 15 } \)

The value of \(\frac { 1 }{ 2! } +\frac { 1 }{ 4! } +\frac { 1 }{ 6! } +....is\)

(a)

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

(b)

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

(c)

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

(d)

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

The value of \(1-\frac { 1 }{ 2 } \left( \frac { 2 }{ 3 } \right) +\frac { 1 }{ 3 } { \left( \frac { 2 }{ 3 } \right) }^{ 2 }-\frac { 1 }{ 4 } { \left( \frac { 2 }{ 3 } \right) }^{ 2 }+....is\)

(a)

\(log\left( \frac { 5 }{ 3 } \right) \)

(b)

\(\frac { 3 }{ 2 } log\left( \frac { 5 }{ 3 } \right) \)

(c)

\(\frac { 5 }{ 3 } log\left( \frac { 5 }{ 3 } \right) \)

(d)

\(\frac { 2 }{ 3 } log\left( \frac { 2 }{ 3 } \right) \)

Expansion of \(log(\sqrt \frac{1+x}{1-x})\) is ______________

(a)

\(x+\frac{x^3}{3}+\frac{x^5}{5}+...\)

(b)

\(1.\frac{x^2}{2}+\frac{x^4}{4}+...\)

(c)

\(1-x+\frac{x^2}{2}+\frac{x^3}{5}+...\)

(d)

\(x-\frac{x^2}{3}+\frac{x^3}{3}+...\)

The value of \(1-\frac{1}{2}(\frac{3}{4})+\frac{1}{3}(\frac{3}{4})^2-\frac{1}{4}(\frac{3}{4})^3+...\)is ______________

(a)

\(\frac{3}{4}log(\frac{7}{4})\)

(b)

\(\frac{4}{3}log(\frac{7}{4})\)

(c)

\(\frac{1}{3}log(\frac{7}{4})\)

(d)

\(\frac{4}{3}log(\frac{4}{7})\)

The slope of the line which makes an angle 45^o with the line 3x- y = -5 are:

(a)

1, -1

(b)

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

(c)

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

(d)

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

Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter 4 + 2\(\sqrt{2}\) is

(a)

x + y + 2 = 0

(b)

x + y - 2 = 0

(c)

\(x+y-\sqrt{2}=0\)

(d)

\(x+y+\sqrt{2}=0\)

The line (p + 2q)x + (p - 3q)y = p - q for different values of p and q passes through the point

(a)

\(\left(\frac{3}{5},\frac{5}{2}\right)\)

(b)

\(\left(\frac{2}{5},\frac{2}{5}\right)\)

(c)

\(\left(\frac{3}{5},\frac{3}{5}\right)\)

(d)

\(\left(\frac{2}{5},\frac{3}{5}\right)\)

If the lines represented by the equations 6x² + 41xy - 7y² = 0 make angles α and β with x-axis, then \(tan \ \alpha\tan \ \beta=\)

(a)

\(-\frac{6}{7}\)

(b)

\(\frac{6}{7}\)

(c)

\(-\frac{7}{6}\)

(d)

\(\frac{7}{6}\)

If the lines x + q = 0, y - 2 = 0 and 3x + 2y + 5 = 0 are concurrent, then the value of q will be ______________

(a)

2

(b)

2

(c)

3

(d)

5

A point equi-distant from the line 4x + 3y + 10 = 0, 5x -12y + 26 = 0 and 7x + 24y - 50 = 0 is ______________

(a)

(1, -1)

(b)

(1, 1)

(c)

(0, 0)

(d)

(0, 1)

If a_ij = \({1\over2}(3i-2j)\) and A = [a_ij]_2x2 is

(a)

\(\begin{bmatrix} {1\over 2}& 2 \\ -{1\over2} & 1 \end{bmatrix}\)

(b)

\(\begin{bmatrix} {1\over 2}& -{1\over2} \\ 2& 1 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 2& 2\\ {1\over 2}& -{1\over2} \end{bmatrix}\)

(d)

\(\begin{bmatrix} -{1\over 2}& {1\over2} \\ 1& 2 \end{bmatrix}\)

If the points (x,−2), (5, 2), (8, 8) are collinear, then x is equal to

(a)

-3

(b)

\({1\over 3}\)

(c)

1

(d)

3