If  and  are in arithmetic progression then the value of x is

a) 0

b) −1

c) 3

d) None of these

If the sum of the series  is a finite number then

a)

b)

c)

d)

The following question has two statements, Statement 1(assertion) and Statement 2 (reason). Each of these questions has 4 alternative choices, only one of which is the correct answer. You have to select one of the codes (a), (b), (c), (d) given below
Statement 1 - for n ε ℕ,
Statement 2 – Geometric mean is greater than Harmonic mean and (AM) (HM) =

a) Statement 1 is false. Statement 2 is true

b) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of Statement 1

c) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of Statement 1

d) Statement 1 is true. Statement 2 is false

If   then  equals

a)

b)

c)

d) None of these

For 2 ≤ r ≤ n,  is equal to

a)

b)

c)

d)

If , then find the values of n and r

For any positive integers m, n (with n ≥ m), prove that
  

A country produces 90% of its food diet. The population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of last year. Assuming that the average food requirement per person remains constant, prove that the country will become self sufficient in food after n years, where n is the smallest integer bigger than or equal to

Prove that

Solve in real numbers the system of equations
 
 

a)

b)

c)

d)

Let a, b, c be distinct positive integers such that
ab + bc +ca ≥ 3k² – 1
where k is a positive integer. Prove that
a³ + b³ + c³ ≥ 3(abc + 3k)

Find minimum of the function f : (0, ∞)³ → ℝ
 

a)

b)

c)

d)

Let <a_n> be a sequence of real numbers such that
 
Prove that

Find min max (a² + b, b² + a)
a, b  ε ℝ

a)

b)

c)

d)

Find all positive integer n for which the equation
nx⁴ +4x +3 = 0
has a real root.

a) n = 1

b) n = 2

c) n = 3

d) n = 4

Find all triplets (x, y, z) of real numbers that are solutions to the system of equations
 
 
 

a) (0, 0, 0)

b)

c) (0, 0, 0),

d) (0, 0, 0),

Let a, b be real numbers such that
9a² + 8ab + 7b² ≤ 6.
Prove that 7a + 5b +12 ab ≤ 9

Let a₁, a₂, .  .  .  . a_n be real numbers such that
a₁ + a₂ + .  .  . . + a_n ≥ n² and
. Prove that
n – 1 ≤ a_k ≤ n + 1 for all k.

If a, b, c are positive numbers prove that
 

If  a₁ + a₂ +.  .  .  . + a_n = n prove that
 

Find the maximum of the function
f (x, y, z) = 5x – 6y + 7z on the ellipsoid
2x² + 3y² + 4z² = 1

a) 2

b) 3

c)

d)

Let f₁, f₂, .  .  . f_n be positive real numbers. Prove that for any real number x₁, x₂, .  .  .  .x_n the quantity
 

is non–negative

Consider the real numbers . Prove that
When does equality hold?