21.

Using principle of mathematical induction, prove that

for all

Using principle of mathematical induction, show that

Using principle of mathematical induction, show that  for all

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)

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