For real numbers x and y, define xRy if x - y + √2 is an irrational number. Then the relation R is __________

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

none of these

The number of students who take both the subjects Mathematics and Chemistry is 70. This represents 10% of the enrollment in Mathematics and 14% of the enrollment in Chemistry. The number of students take at least one of these two subjects, is

(a)

1120

(b)

1130

(c)

1100

(d)

insufficient data

If \(\frac { |x-2| }{ x-2 } \ge 0\), then x belongs to

(a)

\([2,\infty]\)

(b)

\((2,\infty )\)

(c)

\((-\infty,2)\)

(d)

\((-2,\infty )\)

If sin(45 ° + 10°) - sin(45° -10°) = \(\sqrt{2}\)sin x then x is ___________ 

(a)

0^o

(b)

5°

(c)

10°

(d)

15°

If \(\alpha\) and \(\beta\) are two values of θ obtained from the equation a cos θ + b sin θ = c then the value of \(tan(\frac{\alpha+\beta}{2})\) is _______________

(a)

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

(b)

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

(c)

\(\frac{c}{a}\)

(d)

\(\frac{c}{b}\)

The number of ways to average the letters of the word CHEESE are _________

(a)

120

(b)

240

(c)

720

(d)

6

If ⁿP_r = 720, ⁿC_r = 120 then r is _________

(a)

2

(b)

4

(c)

3

(d)

5

The n^th term of the sequence \(\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 8 } ,\frac { 15 }{ 6 } \),......is

(a)

2ⁿ - n - 1

(b)

1 - 2^-n

(c)

2^-n + n - 1

(d)

2^n-1

If 7x² - 8xy +A = 0 represents a pair of perpendicular lines, the A is ______________

(a)

7

(b)

-7

(c)

-8

(d)

8

If the points (2k, k) (k, 2k) and (k, k) enclose a triangle of area 18 sq units, then the centroid of the triangle is ______________

(a)

(8, 8)

(b)

(4, 4)

(c)

(3, 3)

(d)

(2, 2)

If the square of the matrix \(\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is the unit matrix of order 2, then \(\alpha ,\beta \) and \(\gamma\) should satisfy the relation.

(a)

1 + \(\alpha ^2+\beta \gamma=0\)

(b)

1 - \(\alpha ^2-\beta \gamma=0\)

(c)

1 - \(\alpha ^2+\beta \gamma=0\)

(d)

1 + \(\alpha ^2-\beta \gamma=0\)

If A is a square matrix of order 3, then the number of minors in determinant of A are _____________

(a)

3

(b)

21

(c)

9

(d)

27

If \(\overrightarrow{a},\overrightarrow{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

(a)

\(\overrightarrow{a}+\overrightarrow{b}\)

(b)

\({2\overrightarrow{a}-\overrightarrow{b}\over 2}\)

(c)

\({2\overrightarrow{a}+\overrightarrow{b}\over 3}\)

(d)

\({\overrightarrow{a}-\overrightarrow{b}\over 3}\)

\(lim_{\alpha \rightarrow {\pi/4}}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}\) is

(a)

\(\sqrt{2}\)

(b)

\(1\over \sqrt{2}\)

(c)

1

(d)

2

For what values of x is the rate of increase of x³ - 2x² + 3x + 8 is twice the rate of increase of x?

(a)

\(\left( -\frac { 1 }{ 3 } ,-3 \right) \)

(b)

\(\left( \frac { 1 }{ 3 } ,3 \right) \)

(c)

\(\left( -\frac { 1 }{ 3 } ,3 \right) \)

(d)

\(\left( \frac { 1 }{ 3 } ,1 \right) \)

If y = \({1\over4}u^4,u={2\over 3}x^3+5,\) then \({dy\over dx}\) is

(a)

\({1\over 27}x^2(2 x^3+15)^3\)

(b)

\({2\over 27}x(2 x^3+5)^3\)

(c)

\({2\over 27}x^2(2 x^3+15)^3\)

(d)

\(-{2\over 27}x(2 x^3+5)^3\)

\(\int \frac{x^2+\cos ^2 x}{x^2+1} \operatorname{cosec}^2 x d x\) is

(a)

cot x + sin ^-1x + c

(b)

-cot x + tan^-1x + c

(c)

-tan x + cot^-1x + c

(d)

-cot x - tan^-1x + c

\(\int { x } \) sin x dx = -x cos x + a, then a =  

(a)

sin x + c

(b)

cos x + c

(c)

c

(d)

none of these

A number x is chosen at random from the first 100 natural numbers. Let A be the event of numbers which satisfies\({(x-10)(x-50)\over x-30}\ge0\), then P(A) is

(a)

0.20

(b)

0.51

(c)

0.71

(d)

0.70

If P(A\(\cup \)B) = 0.8 and P(A\(\cap \)B) = 0.3 then \(P(\bar { A } )+P(\bar { B } )\) =

(a)

0.3

(b)

0.5

(c)

0.7

(d)

0.9

If X = {1, 2, 3, .. 10} and A = {1, 2, 3, 4, 5}, find the number of sets \(B\subseteq X\) such that A - B = {4}.

Find the value of \(cot(\frac{-15\pi}{4})\).

Write down all the permutations of the vowels A, E, I, O, U in English alphabets taking there at a time starting with A.

Find the general term in the expansion of \({ \left( \frac { 4x }{ 5 } -\frac { 5 }{ 2x } \right) }^{ 9 }\)

The sum of the distance of a moving point from the points (4, 0) and (-4, 0) is always 10 units. Find the equation to the locus of the moving point.

Find a direction ratio and direction cosines of the following vectors \(3\hat{i}+4\hat{j}-6\hat{k}\)

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

Find the derivatives of the following functions using first principle. f(x) = - x² + 2

Integrate the following with respect to x : \({1\over \sqrt{1-25 x^2}}\)

The ratio of the number of boys to the number of girls in a class is 1:2. It is known that the probability of a girl and a boy getting a first class are 0.25 and 0.28 respectively. Find the probability that a student chosen  at random will get first class?

Consider the functions:(i) y = e^x; (ii) y = log_eX.

Evaluate log \({{(\sqrt{\sqrt{625}+11})(\sqrt{64})}\over{\sqrt{\sqrt [ 5 ]{ 3125 }+\sqrt{\sqrt [ 3 ]{ 343 } } }}}\)

What is the length of the arc intercepted by a central angle of measure 41⁰ in a circle radius 10 ft ?

Write the first six terms of the sequences given by a₁ = a₂ = 1, an = a_n-1+ a_n-2 (n ≥ 3)

If the angle between two lines is \(\frac{\pi}{4}\) and slope of one of the line is \(\frac{1}{2}\) find the slope of the other line

Calculate \(lim_{x \rightarrow \infty}{1-x^3\over 3x+2}\)

If x = \(a\sec ^{ 3 }{ \theta }\) and \(y=a\tan ^{ 3 }{ \theta }\) find \(\frac { dy }{ dx }\) at \(\theta =\frac { \pi }{ 3 }\)

Evaluate the following : \(\int \sqrt{(x-3)(5-x)}dx\)

A town has 2 fire engines operating independently. The probability that a fire engine is available when needed is 0.96.
(i) What is the probability that a fire engine is available when needed?
(ii) What is the probability that neither is available when needed?

1. Check the following functions for one-to-oneness and ontoness.
   (i) \(f:N\rightarrow N\) defined by f(n) = n².
   (ii) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by f(n) = n².

2. Find the values of k so that the equation x² = -2x(1 + 3k) + 7(3 + 2k) = 0 has real and equal roots.

1. The Government plans to have a circular zoological park of diameter 8 km. A separate area in the form of a segment formed by a chord of length 4 km is to be allotted exclusively for a veterinary hospital in the park. Find the area of the segment to be allotted for the veterinary hospital.

2. By the principle of mathematical induction, prove that, for all integers n \(\ge\)1, 1 + 2 + 3+....+n = \({n(n+1)\over2}\) .

1. Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

2. Show that the lines are 3x + 2y + 9 = 0 and 12x + 8y - 15 = 0 are paralle llines.

1. Find the equation of the line passing through the point of intersection 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x +4y = 7.

2. Find the value of the product \(\begin{vmatrix} log_364 &log_43 \\ log_38 & log_49 \end{vmatrix}\times \begin{vmatrix} log_23 & log_83 \\ log_34 & log_34 \end{vmatrix}\)

1. Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)

2. Evaluate the following limits :\(lim_{x\rightarrow0}{e^x-e^{-x}\over sin x}\)

1. Differentiate \({ \left( \sin { x } \right) }^{ { \cos { ^{ -1x } } } }\) with respect to 'x'.

2. Integrate the following functions with respect to x : \({1\over \sqrt{1-81x^2}}\)

1. Evaluate \(\int { \frac { { x }^{ 2 }{ tan }^{ -1 }\left( { x }^{ 3 } \right) }{ 1+{ x }^{ 6 } } } \)dx

2. for a loaded die, the probabilities of outcomes are given as under
   P(1) = P(2) = \(\frac { 2 }{ 10 } \), P(3) = P(5) = P(6) = \(\frac { 1 }{ 10 } \) and P(4) = \(\frac { 3 }{ 10 } \)
   The die is thrown 2 times. Let A and B be the events as defined below
   A: Getting same number each time
   B: Getting a total score of 10 or more
   Discuss the independency of the events A and B