KVPY Sample Paper KVPY Stream-SX Model Paper-10

lf \[m=\frac{-1+\sqrt{-\,3}}{2}\] 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)

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

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

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