If the number of terms in the expansion of  is 28, then the sum of the coefficients of all the terms in this expansion is

• 64

• 2187

• 243

• 243

Statement − 1: For every natural number n ≥ 2 

Statement −2: For every natural number n ≥ 2,

• Statement −1 is false, Statement −2 is true

• Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

• Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

• Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

If A =  and I =  , then which one of the following holds for all n ≥ 1, by the principle of mathematical induction

• Aⁿ = n^A – (n – 1)I

• Aⁿ = 2^n-1A – (n – 1)I

• Aⁿ = nA + (n – 1)I

• Aⁿ = nA + (n – 1)I

Let S(K) = 1 +3+5+..... (2K-1) = 3+K². Then which of the following is true?

• S(1) is correct

• Principle of mathematical induction can be used to prove the formula

• S(K) ≠S(K+1)

• S(K) ≠S(K+1)

5.

Maximum sum of coefficient in the expansion of (1 – x sinθ + x² )ⁿ is

• 1

• 2ⁿ

• 3ⁿ

• 3ⁿ

For positive integer n, n³ + 2n is always divisible by

• 3

• 7

• 5

• 6

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8 m/s². The time taken by the particle to move the second metre is

• (√2-1)/2 S

• (√2+1)/2 S

• (1 + √2)S

• (√2-1)S

Prove by induction that for n $\in$ N, n² + n is an even integer (n $\ge$ 1)

A particle is moving in a straight line. At time t, the distance between the particle from its starting point is given by x = t - 6t² + t³. Its acceleration will be zero at

• t = 1 unit time

• t = 2 unit time

• t = 3 unit time

• t = 4 unit time

For each n $\in$ N, 2³ⁿ - 1is divisible by

• 7

• 8

• 6

• 16