Question 1

# If A =  and I =  , then which one of the following holds for all n ≥ 1, by the principle of mathematical induction An = nA – (n – 1)I An = 2n-1A – (n – 1)I An = nA + (n – 1)I An = 2n-1A + (n – 1)I

Solution

A.

An = nA – (n – 1)I

By the principle of mathematical induction (1) is true.

Question 2

## 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 243 729

Solution

D.

729

Clearly, number of terms in the expansion of

Question 3

## 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)

Solution

D.

S(K)⇒ S(K+1)

S(K) = 1 + 3 + 5 + ...... + (2K - 1) = 3 + K2
Put K = 1 in both sides
∴ L.H.S = 1 and R.H.S. = 3 + 1 = 4 ⇒ L.H.S. ≠ R.H.S.
Put (K + 1) on both sides in the place of K L.H.S. = 1 + 3 + 5 + .... + (2K - 1) + (2K + 1)
R.H.S. = 3 + (K + 1)2 = 3 + K2 + 2K + 1
Let L.H.S. = R.H.S.
1 + 3 + 5 + ....... + (2K - 1) + (2K + 1) = 3 + K2 + 2K + 1
⇒ 1 + 3 + 5 + ........ + (2K - 1) = 3 + K2 If S(K) is true, then S(K + 1) is also true. Hence, S(K) ⇒ S(K + 1)

Question 4

## Maximum sum of coefficient in the expansion of (1 – x sinθ + x2 )n is 1 2n 3n 0

Solution

C.

3n

Sum of coefficients in (1 – x sinθ + x2 )n is (1 – sinθ + 1)n
(putting x = 1)
This sum is greatest when sinθ = –1, then maximum sum is 3n .

Question 5

## 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 false.

Solution

C.

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

Hence Statement −2 is not a correct explanation of Statement −1.