Prove that
where  and n is an even integer.

Let n be a positive integer and
(1 + x + x²)ⁿ = a₀ + a₁x + a₂x + a₂x² +  .  .  . + a_2nx²ⁿ then prove that
 

Prove that
 

Use mathematical induction to show that (25)^{n + 1} – 24n + 5735 is divisible by (24)² for all n = 1, 2, .  .  .

The sum of the coefficients of the polynomial (1 + x – 3x²)²¹⁶³ is

The coefficient of x⁹⁹ in the polynomial
(x – 1) (x – 2) .  .  . (x – 100) is

The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰  is

(1 + ax)ⁿ = 1 + 8x + 24x² + .  .  . then a = .  . ., n = .  .  .

Let n be a positive integer. If the coefficient of the 2^nd, 3^rd and 4^th terms in the expansion of (1 + x)ⁿ are in arithmetic progression then n = …..

The sum of the rational terms in the expansion of  is

Let (x, y, z) be points with integer coefficients satisfying the system of homogenous equations
3x – y – z = 0, −3x + z = 0, −3x + 2y + z = 0
then the number of points for which x² + y² + z² ≤ 100 is

The smallest value of k for which both the roots of the equation
x²  8kx + 16 (k² – k + 1) = 0 are real, distinct and have a value of at least 4 is

A value of b for which the equation x² + bx – 1 = 0, x² + x + b = 0 have one root in common is

a)

b)

c)

d)

 then the set  is

Let α, β be roots of x² – 6x – 2 = 0 with α > β. If a_n = αⁿ – βⁿ for n ≥ 1 then the value of  is

a) 1

b) 2

c) 3

d) 4

If (x₀, y₀) be the solution of the equations (2x)^ln2 = (3y)^ln3, 3^lnx = 2^lny, then x₀ is

a)

b)

c)

d) 6

If α and β are roots of the equation x² + 6x − 7 = 0, then  is

a) 6

b) 7

c) 6/7

d) ∞

The minimum value of the sum of real numbers a⁻⁵, a⁻⁴, 3a⁻³, 1, a⁸ and a¹⁰ with a > 0 is

If f (x) = cos (lnx) then f (x) f (y) −   has the value of

a) −1

b)

c) −2

d) None of these

Which of the following functions is periodic?

a) f(x) = x – [x] where [x] denotes the greatest integer less than or equal to the real number x

b) f(x) = sin  x ≠ 0, f(0) = 0

c) f(x) = x cos x

d) None of these

Let f (x) = sin x and g (x) = ln|x|. If the range of the composition

functions fog and gof are R₁ and R₂ respectively, then

a) R₁ = [ u : −1 ≤ u < 1], R₂ = [ v : − < v < 0 ]

b) R₁ = [ u : − < u < 0 ], R₂ = [ v : −1 ≤ v ≤ 0]

c) R₁ = [ u : −1 < u < 1], R₂ = [ v : − < v < 0 ]

d) R₁ = [ u : −1 ≤ u ≤ 1], R₂ = [ v : − < v ≤ 0 ]

Let f (x) be defined for all x > 0 and be continuous. If f (x) satisfies
f  = f (x) – f (y) for all x and y and f (e) = 1 then

a) f (x) is bounded

b) f  → 0 as x → 0

c) x f  → 0 as x → 0

d) f (x) = lnx

If f (x) = 3x – 5 then f ^-1 (x)

a) is given by

b) is given by

c)

d)