NDA 2020 Maths · Mathematics · Complex Numbers and Quadratic Equations · medium

The number of integers n with 1≤ n≤ 20 for which (1+i)^n is a real number is

Single correct option

The number of integers

n

with

1\le n\le 20

for which

(1+i)^n

is a real number is

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Answer: (B) 5

(1+i)^n=2^{n/2}\left(\cos\frac{n\pi}4+i\sin\frac{n\pi}4\right)

is real only when

\sin\frac{n\pi}4=0

, i.e.

n

is a multiple of

4

4,8,12,16,20

, giving

5

values. Counting multiples of

2

(where

(1+i)^2=2i

is purely imaginary) gives the trap value

10

Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake