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Question

Mathematics Question on Complex Numbers and Quadratic Equations

The imaginary part of (12+12i)10\left( \frac{1}{2} + \frac{1}{2}i\right)^{10} is

A

0

B

130\frac{1}{30}

C

131\frac{1}{31}

D

132\frac{1}{32}

Answer

132\frac{1}{32}

Explanation

Solution

We have,
=(12+12i)10=1210(1+i)10= \left(\frac{1}{2} +\frac{1}{2}i\right)^{10} = \frac{1}{2^{10}} \left(1+i\right)^{10}
=1210[(1+i)2]5=1210(2i)5= \frac{1}{2^{10}} \left[\left(1+i\right)^{2}\right]^{5} = \frac{1}{2^{10}} \left(2i\right)^{5}
=1210×25i=132i= \frac{1}{2^{10}} \times2^{5}i = \frac{1}{32} i
Imaginary part =132= \frac{1}{32}

Therefore, the Correct Option is (D): =132= \frac{1}{32}