GATE Overflow Test Series | Quantitative Aptitude | Test 1 | Question: 12

Question

The value of the $\dfrac{(1 - i)^{2023}}{(1 + i)^{2021}},$ where $i = \sqrt{-1}$ is __________

• A. $2$
• B. $-2i$
• C. $-2$
• D. $2i$

Answer 1

$\dfrac{(1 - i)^{2023}}{(1 + i)^{2021}} = \dfrac{(1 - i)^{2023} \times i^{2021}}{(1 + i)^{2021}\times i^{2021}}$

$ = \dfrac{(1 - i)^{2023} \times i^{2021}}{((1 + i)\times i)^{2021}}$

$ = \dfrac{(1 - i)^{2023} \times i^{2021}}{(i + i^{2})^{2021}}$

$ = \dfrac{(1 - i)^{2023} \times i^{2021}}{(i - 1)^{2021}}\qquad [\because i^{2} = -1]$

$ = \dfrac{(1 - i)^{2023} \times i^{2021}}{-(1 - i)^{2021}}$

$ = -(1 - i)^{2} \times i^{2021}$

$ = -(1 - i)^{2} \times i^{1}$

$ = -i(1 - i)^{2}$

$ = -i(1^{2} + i^{2} - 2i)$

$ = -i(1 - 1 - 2i)$

$ = -i \times -2i$

$ = 2i^{2} = 2(-1) = -2$

So, the correct answer is $(C).$

Comments

I feel this should be the answer. Please correct me if wrong.

 

OK GOT MY MISTAKE. cos(1011 pi)=-1 and not 1