Question

If $2x=-1+\sqrt{3}i$, then the value of $(1-x^2+x{)}^6-(1-x+x^2{)}^6$ is

• A. $32$
• B. $-64$
• C. $64$
• D. $0$

Answer 1

The correct option is D

$0$

Explanation for the correct option:

Find the value of given expression:

Given, $2x=-1+\surd 3i$

$\begin{array}{rcl}x& =& \frac{(-1+\surd 3i)}{2}\\ & =& \omega \end{array}$ $\left[\mathbf{\because }\mathit{\omega }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{-}\mathbf{1}\mathbf{}\mathbf{+}\mathbf{}\sqrt{\mathbf{3}}\mathbf{i}}{\mathbf{2}}\right]$

$\therefore$$\begin{array}{rcl}{(1-x^2+x)}^6–{(1-x+x^2)}^6& =& {(1-{\omega }^2+\omega )}^6-(1-\omega +{\omega }^2)\end{array}$

$\begin{array}{rcl}& =& (-2{\omega }^2{)}^6-{(-2\omega )}^6\end{array}$

$\begin{array}{rcl}& =& 2^6{\omega }^{12}-2^6{\omega }^6\end{array}$

$=2^6-2^6$ $\left[\mathbf{\because }{\mathit{\omega }}^{\mathbf{3}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right]$

$=\mathbf{0}$

Hence, Option ‘D’ is Correct.