KVPY Sample Paper KVPY Stream-SX Model Paper-11

Let \[{{z}_{0}}\] be a root of the quadratic equation,\[{{x}^{2}}+x+1=0.\] If \[z=3+6iz_{_{0}}^{81}-3iz_{_{0}}^{93},\] then arg z is equal to:

A) \[\frac{\pi }{4}\]

B) \[\frac{\pi }{3}\]

C) 0

D) \[\frac{\pi }{6}\]

Correct Answer: A

Solution :

\[{{z}_{0}}=\omega \]or \[{{\omega }^{2}}\](where co is a non-real cube root of unity)

\[z=3+6i\,{{(\omega )}^{81}}-3i{{(\omega )}^{93}}\]

\[z=3+3i\]

\[\Rightarrow \] arg \[z=\frac{\pi }{4}.\]