Question

$z$ is a complex number such that $z+\frac{1}{z}=2\cos 3^o$ and |z|=1, then the value of $z^{2000}+\frac{1}{z^{2000}}+1$ is equal to

• A. $0$
• B. $-1$
• C. $\sqrt{3}+1$
• D. $1-\sqrt{3}$

Answer 1

Answer: (A)

$0$

$\begin{array}{c}z+\frac{1}{z}=2\cos 3^0\\ \Rightarrow z=\cos 3^0+i\sin 3^0\Rightarrow e^{i\frac{\pi }{60}}\\ z^{2000}+\frac{1}{z^{2000}}+1=e^{i\frac{2000\pi }{60}}+e^{-i\frac{2000\pi }{60}}+1\\ z^{2000}+\frac{1}{z^{2000}}+1=e^{i\frac{100\pi }{3}}+e^{-i\frac{100\pi }{3}}+1\\ z^{2000}+\frac{1}{z^{2000}}+1=2\cos \left(\frac{100\pi }{3}\right)+1\\ =2\times \frac{-1}{2}+1=0\end{array}$

Hence, option$A$ is correct answer.