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Question

Mathematics Question on Complex Numbers and Quadratic Equations

(1+i)3+(1−i)3{{(1+i)}^{3}}+{{(1-i)}^{3}}

A

11

B

−2-2

C

00

D

−4-4

Answer

−4-4

Explanation

Solution

(1+i)3+(1−i)3=(1+i)2(1+i)+(1−i)2(1−i){{(1+i)}^{3}}+{{(1-i)}^{3}}={{(1+i)}^{2}}(1+i)+{{(1-i)}^{2}}(1-i) =(1+i2+2i)(1+i)+(1+i2−2i) (1−i)=(1+{{i}^{2}}+2i)(1+i)+(1+{{i}^{2}}-2i)\,(1-i) [∵     (a+b)2=a2+b2+2ab (a−b)2=a2+b2−2ab ]\left[ \begin{aligned} & \because \,\,\,\,\,{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\\ & {{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\\ \end{aligned} \right] =2i(1+i)+(−2i)(1−i)=2i(1+i)+(-2i)(1-i) (∵  i2=−1)(\because \,\,{{i}^{2}}=-1) =2i+2i2−2i+2i2=2i+2{{i}^{2}}-2i+2{{i}^{2}} =4i2=−4=4{{i}^{2}}=-4 (∵  i2=−1)(\because \,\,{{i}^{2}}=-1)