If dy/dx = y + 3 > 0 and y(0) = 2, then y(ln2) is equal to

- -13
- 5
- 7
- 2

The solution of the equation d^{2}y/dx^{2} = e^{-2x} is

- e
^{-2x}/4 - 1/4 * e
^{-2x}+ cx^{2}+ d - e
^{-2x}/4 + cx + d - 1/4 * e
^{-4x}+ cx + d

The integral _{0}∫^{π} √(1 + 4 sin^{2}x/2 - 4 sinx/2) dx equals

- 4√3 - 4
- 4√3 - 4 - π/3
- 2π/3 - 4 - 4√3
- π - 4

The integral ∫(1 + x - 1/x)e^{x + 1/x} dx is equal to

- xe
^{x + 1/x}+ c - (x + 1)e
^{x + 1/x}+ c - (x - 1)e
^{x + 1/x}+ c - -xe
^{x + 1/x}+ c

If the integral ∫ (5 tan x / tan x − 2)dx = x + a ln |sin x - 2 cos x| + k, then a is equal to

- -1
- 2
- -2
- 1

If g(x) = _{0}∫^{x} cos4t dt, then g(x + π) equals

- g(x) - g(π)
- g(x) / g(π)
- g(x) + g(π)
- g(x).g(π)

_{0}∫^{π} [cot x] dx, where [.] denotes the greatest integer function, is equal to

- -π/2
- 1
- -1
- π/2

Let f(x) = 4 and f′(x) = 4. Then Lim_{x→2} (xf(2) - 2f(x))/(x-2) is given by

- -4
- 2
- -2
- 3

If f: R → R is a function defined by f(x) = [x] cos((2x - 1)/2)π, where [x] denotes the greatest integer function, then f is

- discontinuous only at x = 0
- discontinuous only at non-zero integral values of x
- continuous only at x = 0
- continuous for every real x

Let y be an implicit function of x defined by x^{2x} - 2x^{x }cot y - 1 = 0. Then y′(1) equals

- -1
- 1
- -log 2
- log 2

The value of p and q for which the function

f(x) = (sin(p + 1)x + sinx)/x, x < 0

f(x) = q, x = 0

f(x) = (√(x + x^{2}) - √x)/x^{3/2}, x > 0

is continuous for all x in R, are:

- p = 1/2 , q = 3/2
- p = -3/2 , q = 1/2
- p = 1/2 , q = -3/2
- p = 5/2 , q = 1/2

Let f(x) be a polynomial function of second degree. If f(1) = f(–1) and a, b, c are in A.P., then f'(a), f'(b) and f'(c) are in

- G.P.
- A.P.
- A.P. - G.P.
- H.P.

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is

- 31
- 29.5
- 28
- 26.5

The sum to infinity of the series 1 + 2/3 + 6/3^{2} + 10/3^{3} + ... is

- 2
- 3
- 4
- 6

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

- √2 + √3
- 2 + √3
- 2 - √3
- 3 + √2

If 100 times the 100^{th} term of an AP with non zero common difference equals the 50 times its 50^{th} term, then the 150^{th} term of this AP is

- 0
- 150
- -150
- 150 times its 50
^{th}term

Fifth term of a GP is 2, then the product of its 9 terms is

- 128
- 256
- 512
- 1024

Sum of infinite number of terms in GP is 20 and sum of their square is 100. The common ratio of GP is

- 1/5
- 3/5
- 8/5
- 5

If the system of linear equations x + 2ay + az = 0, x + 3by + bz = 0, x + 4cy + cz = 0 has a non-zero solution, then a, b, c

- satisfy a + 2b + 3c
- are in G.P
- are in A.P
- are in H.P

The positive integer just greater than (1 + .0001)^{10000} is

- 2
- 3
- 4
- 5

If X = {4^{n} - 3n -1 : n ∈ N} and Y = {9(n - 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to

- Y
- Y - X
- X
- N

The coefficient of x^{7} in the expansion of (1 - x - x^{2} + x^{3})^{6} is

- 132
- -132
- 144
- -144

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

- 6 . 7 .
^{8}C_{4} - 7 .
^{6}C_{4}.^{8}C_{4} - 8 .
^{6}C_{4}.^{7}C_{4} - 6 . 8 .
^{7}C_{4}

If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

- 600
- 601
- 602
- 603

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

- 5! x 6!
- 5! x 4!
- 7! x 5!
- 30

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 are

- 216
- 375
- 400
- 720

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangements is

- at least 750 but less than 1000
- at least 500 but less than 750
- less than 500
- at least 1000

If A and B are square matrices of size n × n such that A^{2} – B^{2} = (A – B)(A + B), then which of the following will be always true?

- either A or B is an identity matrix
- either A or B is a zero matrix
- AB = BA
- A = B

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

- If det A = ± 1, then A
^{-1}exists and all its entries are integers - If det A = ± 1, then A
^{-1}exists and all its entries are non-integers - If det A = ± 1, then A
^{-1}exists and all its entries are not necessarily integers - If det A = ± 1, then A
^{-1}need not exist

Let P and Q be 3 × 3 matrices with P ≠ Q. If P^{3} = Q^{3} and P^{2}Q = Q^{2}P, then determinant of (P^{2} + Q^{2}) is equal to

- 0
- 1
- -1
- -2

Let a,b be real and z be a complex number. If z^{2} + az + b = 0 has two distinct roots on the line Re(z) = 1, then it is necessary that

- |b| = 1
- b ∈ (0,1)
- b ∈ (1,∞)
- b ∈ (-1,0)

Let z_{1} and z_{2} be two roots of the equation z^{2} + az + b = 0, z being complex further, assume that the origin, z_{1} and z_{2} form an equilateral triangle, then

- a
^{2}= 4b - a
^{2}= 3b - a
^{2}= 2b - a
^{2}= b

If z^{2} + z + 1 = 0, where z is a complex number, then the value of

(z + 1/z)^{2} + (z^{2} + 1/z^{2})^{2} + ... + (z^{6} + 1/z^{6})^{2}

- 6
- 12
- 18
- 54

The locus of the centre of a circle which touches the circle |z - z_{1}| = a and |z - z_{2}| = b externaly (z, z_{1} & z_{2} are complex numbers) will be

- a hyperbola
- an ellipse
- a circle
- a straight line

The vectors a and b are not perpendicular and c and d are two vectors satisfying: b x c = b x d and a.d = 0. Then the vector d is equal to

- b - (b.c/a.d)c
- c + (a.c/a.b)b
- c - (a.c/a.b)b
- b - (b.c/a.d)c

If the vectors a = i - j + 2k, b = 2i + 4j + k and c = ai + j + bk are mutually orthogonal, then (a,b) =

- (-2,3)
- (3,-2)
- (-3,2)
- (2,-3)

If p and q are the roots of the equation x^{2} + px + q = 0, then

- p = 1, q = -2
- p = -2, q = 1
- p = -2, q = 0
- p = 0, q = 1

If a and b are the roots of the equation x^{2} - x + 1 = 0, then a^{2009} + b^{2009} is equal to

- -2
- 2
- -1
- 1

The equation e^{sinx} – e^{-sinx} – 4 = 0 has

- Infinite number of real roots
- No real roots
- Exactly one real root
- Exactly four real roots

The domain of sin^{-1}[log_{3}(x/3)] is

- [-9, -1]
- [-9, 1]
- [-1,9]
- [1, 9]

Let W denote the words in the English dictionary. Define the relation R by:

R = {(x, y) ε W × W | the words x and y have at least one letter in common}. Then R is

- reflexive, symmetric and not transitive
- reflexive, symmetric and transitive
- not reflexive, symmetric and transitive
- reflexive, not symmetric and transitive

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

- (-∞,1/3] ; 3x
^{2}- 2x + 1 - [2,∞) ; 2x
^{3}- 3x^{2}-12x + 6 - (-∞,∞) ; x
^{3}- 3x^{2}+ 3x + 3 - (-∞,-4] ; x
^{3}+ 6x^{2}+ 6

If |z - 4/z| = 2, then the maximum value of |Z| is equal to

- √3 + 1
- √5 + 1
- 2 + √2
- 2

If w (≠1) is a cube root of unity and (1 + w)^{7} = A + Bw. Then (A, B) equals to

- (1, 1)
- (1, 0)
- (0, 1)
- (-1, 1)

The Number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals

- 0
- 1
- 2
- infinity

Let a = j - k and c = i - j - k. Then the vector b satisfying a x b + c = 0 and a.b = 3 is

- i - j - 2k
- i + j - 2k
- 2i - j + 2k
- -i + j - 2k

If |a| = 4, |b| = 2 and the angle between a and b is π/6, then (a × b)^{2} is equal to

- 14
- 16
- 36
- 48

a, b and c are 3 vectors, such that a + b + c = 0, |a| = 1, |b| = 2|c|, then a.b + b.c + c.a is equal to

- 1
- 0
- -7
- 7

Let a and b be two unit vectors. If the vectors c = a + 2b and d = 5a - 4b are perpendicular to each other, then the angle between a and b is

- π/4
- π/6
- π/2
- π/3

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

- x
^{2}+ 18x + 16 = 0 - x
^{2}- 18x - 16 = 0 - x
^{2}- 18x + 16 = 0 - x
^{2}+ 18x - 16 = 0