The sum to infinity of the series    is

a) 3

b) 4

c) 6

d) 2

Let  for  and , where in the composition, f occurs n times. Then  is equal to

a)

b)

c)

d)

If

Then

a)

b)

c)

d)

Let f(x) be differentiable for all x. If f (1) = − 2 and  for x [1, 6] then

a) f(6)=5

b) f(6) < 5

c) f(6) < 8

d) f(6) ≥ 8

Let f(x) be a polynomial function of second degree. If f (1) = f (−1) and a, b, c are in arithmetic progression then  and  are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) Arithmetic – Geometric Progression

If  where [x] denotes the integral part of x then

a) f is one-one

b) f is not one-one and non-constant function

c) f is a constant function

d) None of the above

Let g(x) = 1 + x –[x] and  

then for all x, f(g(x)) is equal to

a) x

b) 1

c) f(x)

d) g(x)

Prove that  is an irrational number.

Prove that  is an irrational number.

Solve the inequality

|x² – 7x + 12| > x² – 7x + 12

a) x ε (3, 4)

b) x ε (0, 4)

c) x ε (0, 3)

d) x ε (0, 3) ∪ (3, 4)

Determine the values of x satisfying the equation
 

a) – 1 ≤ x ≤ 1

b) – 1 ≤ x ≤ 0 and x > 0

c) x ≥ 1 and x < − 1

d) – 1 < x ≤ 1

Find the roots of the equation
|sinx| = sinx + 1

a)

b)

c) ; n = 0, ±1, ±2, ...

d) ; n = 0, ±1, ±2, ...

Let a, b, c be any real numbers. Suppose that there exist real numbers x, y, z not all zero such that

x = cy + bz

y = az + cx and

z = bx + ay

then is equal to

a) 1

b) 2

c) –1

d) 0

Let A be a square matrix whose entries are integers. Which of the following is true?

a) If detA = ± 1 then  need not exist

b) If detA = ± 1 then  exists but all its entries are not necessarily integers.

c) If detA ≠ ± 1 then  exists and its entries are non-integers.

d) If detA = ± 1 then  exists and its entries are integers

If  where x ≠ 0, y ≠ 0 then D is

a) Divisible by neither x nor y

b) Divisible by both x and y

c) Divisible by x but not y

d) Divisible by y but not x

Let . If  then |α| equals

a)

b)

c)

d) 5

If A and B are square matrices of size  such that
then which of the following will always be true?

a) AB = BA

b) Either A or B is a zero matrix

c) Either A or B is a identity matrix

d) There cannot exist any B such that AB = BA

Let  then

a) There exist more than one but finite numbers of B’s such that AB = BA

b) There exists exactly one B such that AB = BA

c) There exist infinitely many B’s such that AB = BA

d) There cannot exist any B such that AB = BA

If  then the inverse of A is

a) I – A

b) A – I

c) A

d) A + I

If
 
Then f(x) is a polynomial of degree

a) 2

b) 3

c) 0

d) 1

The system of equations
αx + y + z =α–1
x + αy + z = α–1
x + y + αz = α–1
has no solution if α is

a) 1

b) Not –2

c) Either –2 or 1

d) –2

If  , .  .  . are in geometric progression then the determinant Δ is equal to

a) 2

b) 4

c) 0

d) 1

Let .

The only correct statement about matrix A is

a) A is a zero matrix

b) A =  I where I is  a unit matrix

c) does not exist

d) A² = I