Question

If a complex number $z$ has modulus $1$ and argument $\frac{\pi }{3}$, then $z^2+z$

• A. is purely imaginary
• B. has modulus $\sqrt{3}$
• C. lies on the imaginary axis
• D. None of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A is purely imaginary
B has modulus $\sqrt{3}$
C lies on the imaginary axis

$z^2=e^{i\frac{2\pi }{3}}=\frac{-1}{2}+i\frac{\sqrt{3}}{2}$
$z=e^{i\frac{\pi }{3}}=\frac{1}{2}+i\frac{\sqrt{3}}{2}$
$z^2+z$
$=2\left(i\frac{\sqrt{3}}{2}\right)$
$=i\sqrt{3}$

Hence, options A, B and C are correct