Question

If $(1-i{)}^n=2^n$ , then n is

• A. 1
• B. 0
• C. -1
• D. None of these

Answer 1

The correct option is B

0

We know that if two complex numbers are equal ,

their moduli must also be equal, therefore from (i), we have

| $(1-i{)}^n$ | = |$(2{)}^n$| ⇒ $|(1-i){|}^n$ = $|2{|}^n$ , ( ∵ $2^n$ > 0)

⇒ ${\left[\sqrt{1^2+(-1{)}^2}\right]}^n$ = $2^n$ ⇒ ($\sqrt{2}{)}^n$) = $2^n$
⇒ $2^{\frac{n}{2}}$ = $2^n$ ⇒ $\frac{n}{2}$ ⇒ n =0

Trick : By inspection, $(1-i{)}^0$ = $2^0$ ⇒ 1 = 1