The fixed nonzero complex number $a$ has modulus $R$ and argument $\alpha$, and $a^*$ denote

Question

The fixed nonzero complex number $a$ has modulus $R$ and argument $\alpha$, and $a^*$ denotes its complex conjugate. Given that $q=\dfrac{a^3}{a^*}$, express $q$ in trigonometric form.[3] If one of the sixth roots of $q$ is purely imaginary, determine the least positive value of $\alpha$.[3]

Accepted Answer

$q=R^2(\cos4\alpha+i\sin4\alpha)$, with least positive $\alpha=\pi/4$.

2009 CJC P1 Q8 (b)

2009 CJC P1 Q8 (b)

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