For n ε N, 10^{n -2} ≥ 81n if

a) n  >  5

b) n  ≥  5

c) n  <  5

d) n  >  8

Let S_n =  +  +  . . . . . . . . . +  , n = 1, 2, 3 . . . . . .then S_n is not greater than

a)

b)

c) 2

d) 4

The smallest value of x satisfying the equation

= 2 is

a)

b)

c)

d)

The question has four alternative choices, only one of which is the correct answer. Select one of them which is correct

Statement I     for n ε N, 2ⁿ  > 1 + n
Statement II   GM > HM and (AM) (HM) = (GM)²

a) Statement I is false, Statement II is correct.

b) Statement I is true, Statement II is true. Statement II is a correct explanation for the statement.

c) Statement I is true, Statement II is true. Statement II is not a correct explanation for the statement.

d) Statement I is true, Statement II is false.

The minimum number of terms from the beginning of the series 20 + 22  + 25 + ……. so that the sum may exceed 1568, is

a) 25

b) 27

c) 29

d) 28

The question has four distinct choices, only one of which is the correct answer. Select one of them which is correct

Statement I: If both roots of the equation 4x² – 2x + a = 0, a ε ℝ lie in the interval (-1, 1), then 2 < a ≤ .
Statement II : If f(x) = 4x² – 2x + a then D ≥ 0, f(-1) > 0, f(1) > 0 ⇒ 2 < a ≤ 

a) Statement I is false, Statement II is true.

b) Statement I is true, Statement II is true. Statement II is a correct explanation for the statement I.

c) Statement I is true, Statement II is true. Statement II is not a correct explanation for the statement I.

d) Statement I is true, Statement II is false.

The question has four distinct choices, only one of which is the correct answer. Select one of them which is correct
Statement I: If a, b, c are distinct and x, y, z are not all zeros and ax + by + cz = 0, bx + cy + az = 0, cx + ay +bz = 0 then a + b + c = 0
Statement II: a² + b² + c² > ab + bc + ca if a, b, c are distinct

a) Statement I is false, Statement II is true.

b) Statement I is true, Statement II is true. Statement II is a correct explanation for the statement I.

c) Statement I is true, Statement II is true. Statement II is not a correct explanation for the statement I.

d) Statement I is true, Statement II is false.

The value of the sum of the series 3 ⁿC₀ – 8 ⁿC₁ + 13 ⁿC₂ – 18 ⁿC₃ +  .  .  .  .  . + ( n + 1 ) terms is

a) 0

b) 3ⁿ

c) 5ⁿ

d) None of these

The sum of first n terms of the series  +  +  +  + .  .  .  . is equal to

a) 2ⁿ – n + 1

b) 1 – 2^-n

c) n + 2^-n – 1

d) 2ⁿ – 1

The value of x in the given equation 4^x – 3^{x – ½} = 3^{x + ½} – 2^{2x – 1} is

a)

b)

c)

d)

The solution set of the inequation
 +1 < 0  is

a) (, 2) ∪ (1, )

b) [1, 2]

c) (2, 1)

d) ()

The sum of n terms of three Arithmetic Progressions whose first term is a and whose common difference are 1, 2, 3 are S₁, S₂, S₃ respectively. The true relation is

a) S₁ + S₃ = S₂

b) S₁ + S₃ = 2S₂

c) S₁ + S₂ = 2S₃

d) S₁ + S₂ = S₃

The Arithmetic mean, Harmonic mean and Geometric mean between two numbers are , 15 and 12, but not necessarily in the same order. Then harmonic mean, geometric mean and arithmetic mean respectively are

a) 15, 12,

b) , 12, 15

c) 12, 15,

d) , 15, 12

If ( 1 + x )ⁿ = ⁿC_r x^r, then
( 1 +  ) . ( 1 +  ) .  .  .  . . ( 1 +  ) is equal to

a)

b)

c)

d)

If x₁, x₂, …. , x_n are in Arithmetic Progression whose common difference is α then the value of sin α { sec x₁ sec x₂ + sec x₂ sec x₃ + ….. + sec x_{n - 1} sec x_n } is equal to

a)  

b)

c)

d)

The number of real values of the parameter k for which ()² –  +  = 0 with real coefficients will have exactly one solution is

a) 2

b) 1

c) 4

d) None of these

If the (m + 1)th, (n + 1)th and the (r + 1)th term of an arithmetic progression are in geometric progression, and m, n, r are in harmonic progression then the value of the ratio of the common difference to the first term of the arithmetic progression is

a)

b)

c)

d)

The coefficient of x⁵³ in the following expansion
 ¹⁰⁰C_m (x – 3)^{100 – m} 2^m  is

a) ¹⁰⁰C₄₇

b) ¹⁰⁰C₅₃

c) – ¹⁰⁰C₅₃

d) – ¹⁰⁰C₁₀₀

If x² + px + 1 is a factor of the expression ax³ + bx + c then

a) a² + c² =  ab

b) a² – c² = ab

c) a² – c² =  ab

d) none of these

The 14^th term from the end in the expansion of ¹⁷ is

a) ¹⁷C₅ x⁶⁵

b) ¹⁷C₆ ¹¹ y³

c) ¹⁷C₄ x^13/2y²

d) None of these

Let A be a 2 x 2 matrix with real entries. Let I be the 2 x 2 identity matrix. Denote by tr (A), the sum of the diagonal entries of A. Assume that A² = I
Statement I : If A ≠ I and A ≠ – I then det (A) = –1
Statement II : If A ≠ I and A is ≠ – I the tr (A) = 0

a) Statement I is false. Statement II is true.

b) Statement I is true. Statement II is true. Statement II is the correct explanation for statement.

c) Statement I is true. Statement II is true. Statement II is not the correct explanation for statement I.

d) Statement I is true. Statement II is false.

If a, b, c are real numbers, suppose that there are real numbers x, y, z not all zero such that x = cy +bz, y = az + cx and z = bx + ay then a² + b² +c² + 2abc is equal to

a) 1

b) 2

c) 1

d) 0

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

a) If det (A) = ±1 then A^-1 need not exist

b) If det (A) = ± 1 then A^-1 exists but all its entries are not necessarily integers

c) If det (A) = ± 1 then A^-1 exists and all its entries are non - integers

d) If det (A) = ± 1 then A^-1 exists and all its entries are integers