KVPY Sample Paper KVPY Stream-SX Model Paper-24

If \[m=\frac{-\,1+\sqrt{-\,3}}{2}\] then the expression \[{{(1-m)}^{2}}\,\,{{(m-{{m}^{2}})}^{2}}\,\,{{(1-{{m}^{2}})}^{2}}\] simplifies to -

A) An imaginary number

B) A prime number

C) A positive integer

D) A negative integer

Correct Answer: D

Solution :

Note that \[m=\omega \](cube root of unity)

\[{{(1-\omega )}^{2}}\,\,{{(\omega -{{\omega }^{2}})}^{2}}\,\,{{(1-{{\omega }^{2}})}^{2}}\]

\[=(1+{{\omega }^{2}}-2\omega )\,\,({{\omega }^{2}}+{{\omega }^{4}}-2)\,\,(1+{{\omega }^{4}}-2{{\omega }^{2}})\]

\[=(-\,3\omega )\,\,(-\,3)\,\,(-\,3{{\omega }^{2}})=-\,27\]