Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
The expression $\frac{(1+i)^n}{(1-i)^{n-2}}$ equals
A
$-i^{n+1}$
B
$i ^{ n +1}$
✓
$-2 i ^{ n +1}$
D
1
✓
Answer
Correct option: C.
$-2 i ^{ n +1}$
(C) $\frac{(1+ i )^{ n }}{(1- i )^{ n -2}}=\frac{(1+ i )^{ n }}{2^{ n -2}} \cdot(1+ i )^{ n -2} \ldots\left[\because(1- i )=\frac{2}{(1+ i )}\right]$ $=\frac{(1+i)^{2 n-2}}{2^{n-2}}$ $=\frac{(1+ i )^{2( n -1)}}{2^{ n -2}}$ $=\frac{\left(1+ i ^2+2 i \right)^{ n -1}}{2^{ n -2}}$ $=\frac{(2 i )^{ n -1}}{2^{ n -2}}$ $=2 i ^{ n -1}$ $=2 i ^{ n -1} \cdot i ^2(-1)$ $=-2 i ^{ n +1}$
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