Question

If fourth root of unity are $z_{1,}{z}_{2,}{z}_{3,}{z}_{4,}$, then $z_1^2+z_2^2+z_3^2+z_4^2=$

• A. $1$
• B. $0$
• C. $i$
• D. None of these

Answer 1

Answer: (B)

$0$

$\because$ Square roots of unity are $1$ and $-1$

$\therefore$ Fourth roots of unity are as square roots of $1$ and $-1$ i.e., $1,-1,i,-i$

So, we have $z_1=1,z_2=-1,z_3=i,z_4=-i$

Then, $z_1^2+z_2^2+z_3^2+z_4^2={\left(1\right)}^2+{(-1)}^2+{\left(i\right)}^2+{(-i)}^2=1+1+(-1)+(-1)=2-2=0$