Which one of the following is not a singleton set?

(a)

A = {x : 3x - 5 = 0, x ∈ Q}

(b)

B = {| x | = 1 / x ∈ Z}

(c)

{x : x³ - 1 = 0, x ∈ R}

(d)

{x : 30x = 60, x ∈ N}

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

(a)

16

(b)

18

(c)

9

(d)

12

If ABCD is a cyclic quadrilateral then cos A + cos B + cos C + cos D = _______________

(a)

1

(b)

-1

(c)

0

(d)

None

In ²ⁿC₃ : ⁿC₃ = 11 : 1 then n is

(a)

5

(b)

6

(c)

11

(d)

7

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 first and last term of an A. P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be ______________

(a)

5

(b)

6

(c)

7

(d)

8

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

The equation of a line which makes an angle of 135° with positive direction of x-axis and passes through the point (1, 1) is ______________

(a)

x+y=2

(b)

x-y=0

(c)

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

(d)

x-3y=0

If \(\begin{vmatrix}2a & x_1 &y_1 \\ 2b & x_2 & y_2 \\ 2c & x_3 &y_3 \end{vmatrix}={abc\over 2}\neq 0,\) then the area of the triangle whose vertices are \(\begin{pmatrix} {x_1\over a}, {y_1\over a} \end{pmatrix}\), \(\begin{pmatrix} {x_2\over b}, {y_2\over b} \end{pmatrix}\), \(\begin{pmatrix} {x_3\over c}, {y_3\over c} \end{pmatrix}\) is

(a)

\({1\over 4}\)

(b)

\({1\over 4} abc\)

(c)

\({1\over 8}\)

(d)

\({1\over 8}abc\)

The product of any matrix by the scalar_________is the null matrix.

(a)

1

(b)

0

(c)

I

(d)

matrix itself

The vector in the direction of the vector\(\hat{i}-2\hat{j}+2\hat{k}\) that has magnitude 9 is _________ .

(a)

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

(b)

\(\frac { \hat { i } -2\hat { j } +2\hat { k } }{ 3 } \)

(c)

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

(d)

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

Find the odd one out of the following

(a)

matrix multiplication

(b)

vector cross product

(c)

Subtraction

(d)

Matrix Addition

At x \(={3\over 2}\) the function \(f(x)={|2x-3|\over 2x-3}\) is

(a)

continuous

(b)

discontinuous

(c)

differentiable

(d)

non-zero

The points of discontinuity of the function \(\frac { { x }^{ 2 }+6x+8\quad }{ { x }^{ 2 }-5x+6\quad } is\)

(a)

3,2

(b)

3,-2

(c)

-3,2

(d)

-3,-2

If f(x) = \(\left\{\begin{matrix} x+2& -1<x<3\\ 5,& x=3\\ 8-x,& x>3\\ \end{matrix}\right.\), then at x = 3, f'(x) is:

(a)

1

(b)

-1

(c)

0

(d)

does not exist

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\sqrt { \sin { x+y } } \quad then\quad \frac { dy }{ dx }\) is _________

(a)

\(\frac { \sin { x } }{ 2y-1 } \)

(b)

\(\frac { \sin { x } }{ 1-2y } \)

(c)

\(\frac { \cos { x } }{ 1-2y } \)

(d)

\(\frac { \cos { x } }{ 2y-1 } \)

\(\int \frac{1}{x \sqrt{(\log x)^2-5}} d x\) is

(a)

\(log|x+\sqrt{x^2-5}|+c\)

(b)

\(log|logx+\sqrt{logx-5}|+c\)

(c)

\(log|logx+\sqrt{(logx)^2-5}|+c\)

(d)

\(log|logx-\sqrt{(logx)^2-5}|+c\)

\(\int { \frac { sin\sqrt { x } }{ x } } \) dx = ________ +c.

(a)

2 cos \(\sqrt { x } \)

(b)

2 sin \(\sqrt { x } \)

(c)

-2 sin \(\sqrt { x } \)

(d)

-2 cos \(\sqrt { x } \)

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 flash light has 8 batteries out of which 3 are dead. If 2 batteries are selected without replacement and tested, the probability that both are dead is

(a)

\(\frac { 3 }{ 28 } \)

(b)

\(\frac { 1 }{ 14 } \)

(c)

\(\frac { 9 }{ 64 } \)

(d)

\(\frac { 33 }{ 56 } \)

Evaluate \(\left( \left[ (256)^{ \frac { -1 }{ 2 } } \right] ^{ \frac { -1 }{ 4 } } \right) ^{ 3 }\)

There are 3 types of toy car and 2 types of toy train available in a shop. Find the number of ways a baby can buy a toy car and a toy train?

How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7 if no digit is repeated?

Find the n^th term of the series 3 - 6 + 9 -12 + ...

The length L (in cm) of a copper rod is a linear function of its Celsius temperature C. In an experiment if L = 124.942 when C = 20 and. L = 125.134 when C = 110, express L in terms of C.

Find x, y, z and w such that \(\begin{bmatrix} x-y & 2z+w \\ 2x-y & 2x+w \end{bmatrix}=\begin{bmatrix} 5 & 3 \\ 12 & 15 \end{bmatrix}\)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1+x } +\sqrt { 1-x } }{ 1+x } } \)

Find the derivation : sin 5 + log₁₀ x + 2 sec x

Integrate the following with respect to x : \({1\over cos^2 \ x}\)

If P(\(\bar { A } \)) = 0.6 P(B) = 0.7 and \(P\left( \frac { B }{ A } \right) =0.4\) , then find \(P\left( \frac { A }{ B } \right) \)and \(P(A\cup B)\)

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is reflexive

Simplify : \(\frac { 1 }{ 2+\sqrt { 3 } } +\frac { 3 }{ 4-\sqrt { 5 } } +\frac { 6 }{ 7-\sqrt { 8 } } \)

The angles of a triangle ABC, are in arithmetic progression and if b:c = \(\sqrt { 3 } :\sqrt { 2 } \) , find \(\angle A.\)

Evaluate \(\frac { (2n)! }{ n! } \)

Prove that \({ C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+{ C }_{ 2 }^{ 2 }+...=\frac { (2n)! }{ (n)! } \)

A straight rod of the length 6 units, slides with its ends A and B always on the x and y axes respectively. If O is the origin, then find the locus of the centroid of ΔOAB.

If \(\overrightarrow{a},\overrightarrow{b}\) are unit vectors and \(\theta\) is the angle between them, show that \(sin {\theta \over 2}={1\over2}|\overrightarrow{a}-\overrightarrow{b}|\)

Evaluate \(\lim _{ x\rightarrow 1 }{ \frac { (2x-3)\sqrt { x } -1 }{ { 2x }^{ 2 }+x-3 } } \)

Evaluate the integrate \(\cfrac { 1 }{ 7-(4x+1)^{ 2 } } \)

A and B are two events such that P(A) \(\neq \) 0. Find P(B/A) if (i) A is a subset of B (ii) A\(\cap \)B = \(\phi \)

1. Eliminate \(\theta\) from the equation a sec \(\theta\) - c tan \(\theta\) = b and b sec \(\theta\)  + d tan \(\theta\) = C

2. Out of 10 outstanding students in a school there are 6 girls and 4 boys. A team of 4 students is selected at random for a quiz programme. Find the probability that there are atleast two girls.

1. Prove that \(\frac { (2n)! }{ n! } \) = 2ⁿ (1.3.5...(2n - 1)).

2. Integrate the following with respect to x : \({x+2\over \sqrt{x^2-1}}\)

1. Determine the region in the Plane determined by the inequalities x+y≤9,y>x,x≥0

2. Let a, b, c denote the sides BC, CA and AB respectively of \(\triangle\)ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| \) = 0 then find the value of sin²A + sin²B + sin²C.

1. Using principle of mathematical induction, prove that 41ⁿ -14ⁿ is a multiple of 27.

2. Find the derivatives of the following : y = x^cosx

1. Prove that \(log_{10}2+16log_{10}\frac { 16 }{ 15 } +12log_{10}\frac { 25 }{ 24 } +7log_{10}\frac { 81 }{ 80 } =1\)

2. Prove that \(\sqrt { \frac { 1-x }{ 1+x } } \) is approximately equal to 1 - x + \(\frac{x^2}{2}\) when x is very small.

1. If f:R \(\rightarrow\) R is defined by f(x) = 3x - 5, prove that f is a bijection and find its inverse.

2. Find the distance
   between two parallel lines 3x + 4y = 12 and 6x + 8y + 1 = 0.

1. If \(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\)are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is \({1\over 2}|\overrightarrow{a}\times \overrightarrow{b}+ \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|\). Also deduce the condition for collinearity of the points A, B, and C.

2. Determine k, so that  \(f\left( x \right) =\begin{cases} k{ x }^{ 2 },\quad x\le 2 \\ 3,\quad x>2 \end{cases}\) is continuous.