Question

If $\alpha$ is the fifth root of unity, then

• A. $|1+\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4|=0$
• B. $|1+\alpha +{\alpha }^2+{\alpha }^3|=1$
• C. $|1+\alpha +{\alpha }^2|=2\cos \frac{\pi }{5}$
• D. $|1+\alpha |=2\cos \frac{\pi }{10}$

Answer 1

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

The correct options are
A $|1+\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4|=0$
B $|1+\alpha +{\alpha }^2|=2\cos \frac{\pi }{5}$
C $|1+\alpha +{\alpha }^2+{\alpha }^3|=1$

We have, $\alpha =\cos \frac{2\pi }{5}+i\sin \frac{2\pi }{5}$ and $1+\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4=0$

$\therefore |1+\alpha +{\alpha }^2+{\alpha }^3|=|-{\alpha }^4|={\left|\alpha \right|}^4=1$

Also, $|1+\alpha +a^2|=|-{\alpha }^3(1+\alpha )=(1+\alpha )|$ ...(1)

$=|1+\cos \frac{2\pi }{5}+i\sin \frac{2\pi }{5}|=|2+\cos \frac{\pi }{5}(\cos \frac{\pi }{5}+i\sin \frac{2\pi }{5})|=2\cos \frac{\pi }{5}$

Again, from (i), $|1+\alpha |=|1+\alpha +{\alpha }^2|=2\cos \frac{\pi }{5}.$