If S₁, S₂, .  .  .  .,S_n are the sums of infinite geometric series whose first terms are 1, 2, 3,   .  .  ., n and whose common ratios are  respectively, then find the value of

The real numbers x₁, x₂, x₃ satisfying the equation x³ – x² + βx + γ = 0 are in Arithmetic Progression. Find the interval in which β and γ lie.

Let a₁, a₂, … a_n be positive real numbers in Geometric Progression. For each n let A_n, G_n, H_n be respectively the arithmetic mean, geometric mean and harmonic mean of a₁, a₂, .  .  .  ., a_n. Find the expressions for the Geometric mean of G₁, G₂, .  .  .  .G_n in terms of A₁, A₂, .  .  .  .,A_n; H₁, H₂, .  .  .  .H_n

Let a, b are real positive numbers. If a, A₁, A₂, b are in Arithmetic Progression, a, G₁, G₂, b are in Geometric Progression and a, H₁, H₂, b are in Harmonic Progression show that
 

If total number of runs scored in n matches is
 where n > 1 and the runs scored in the k^th match are given by k.2^{n + 1 – k}  where 1 ≤ k ≤ n. Find n.

If three complex numbers are in Arithmetic Progression, then they lie on a circle in a complex plane.

a) True

b) False

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

In a triangle ABC if cotA, cotB, cotC are in Arithmetic Progression then a, b, c are in .  .  .  .  . Progression.

For any odd integer n ≥ 1,
n³ – (n – 1)³ + .  .  . + (−)^{n – 1} 1³ = .  .  .

Let f (x) = |x – 1| then

a) f (x²) = |f (x)|²

b) f (x + y) = f (x) + f (y)

c) f () = |f (x)|

d) None of these

Let f(x) = (x + 1)² – 1, x ≥ −1 then the set {x : f(x) = f^-1(x)} is

a)

b) { 0, 1, −1}

c) {0, −1}

d) Ф

The function
f(x) =|px – q| + r |x|, x ε (−, )
where p > 0, q > 0, r > 0 assumes minimum value on one point if

a) p ≠ q

b) r = q

c) r ≠ p

d) r = p = q

If f (x) =  for every real x then the minimum value of f

a) does not exist because f is unbounded

b) is not attained even though f is bounded

c) is equal to 1

d) is equal to −1

If the function f : [ 1,  ) → [ 1,  ) is defined by f (x) = 2^{x(x – 1)} then
f ^-1(x) is

a)

b)  ()

c)  ()

d)

Let f : R → R be any function defined g : R → R by g (x) = |f (x)| for all x. Then g is

a) onto if f is onto

b) one to one if f is one to one

c) continuous if f is continuous

d) differentiable if f is differentiable

If f : [ 1,  → [ 2, ] is given by f (x) = x +  then ( x ) is given by

a)

b)

c)

d) 1 +

Let f ( x ) = , x ≠ 1 then for what value of a is f ( f (x)) = x

a)

b)

c) 1

d) 1

Suppose f (x) = (x + 1)² for x ≥ . If g (x) is the function whose graph is the reflection of the graph of f (x) with respect to the line y = x then g (x) equals

a) ,  0

b)

c)

d)

The function of f : R → R be defined by f (x) = 2x + sinx for x ε R . Then f is

a) one-one and onto

b) one-one but not onto

c) onto but not one-one

d) neither one-one nor onto

If f : [ 0,  )  [ 0,  ) and f (x) =  then f is

a) one-one and onto

b) one-one but not onto

c) onto but not one-one

d) neither one-one nor onto

If f (x) = sinx + cosx, g (x) = x² – 1 then g ( f (x)) is invertible in the domain

a)

b)

c)

d)

f (x) =
and g (x) =
 

a) neither one-one nor onto

b) one-one and onto

c) one-one and into

d) many one and onto

If X and Y are two sets and f : X  Y
If { f (c) = y, c ⊂ x, y ⊂ Y } then the true statement is

a)

b)

c) , a ⊂ X

d)