Question

Consider a complex number $z=\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right)$ and a set $A=\left\{z_k\mid z_k=z^k,k=0,1,2,3,\dots \dots 9\right\}$, then

• A. all elements of A lie on a unit circle centred at origin.
• B. $\arg \left(z_p\right)-\arg \left(z_q\right)=\frac{2\pi }{5}$ for exactly 8 ordered pairs of $\left(z_p,z_q\right)$ where $z_p,z_q\in A$ and $\arg z\in [0,2\pi )$.
• C. number of elements of $A$ for which $\mathrm{Re}\left(z_k\right)>\sin 18^{\circ }$ is 3 .
• D. number of elements of A for which $\mathrm{Im}\left(z_k\right)+\sin 36^{\circ }>0$ is 6 .

Answer 1

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

(A) (A) is obivious
(B) $\left(z_2,z_0\right)\left(z_3,z_1\right)\left(z_4,z_2\right)\left(z_5,z_3\right),\left(z_6,z_4\right)\left(z_7,z_5\right)\left(z_8,z_6\right)\left(z_9,z_7\right)$ 8 ordered pairs

(C) $z_0,z_1,z_9$ are 3 elements of $A$ for which $\mathrm{Re}(z)>\sin 18^{\circ }$.
(D) $\mathrm{Im}\left(z_k\right)>-\sin 36^{\circ }\Rightarrow z_0,z_1,z_2,z_3,z_4,z_5$ are required elements of