Question

Let $z$ be a complex number such that the imaginary part of $z$ is non zero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

• A. $-1$
• B. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $\frac{3}{4}$

Answer 1

The correct option is D $\frac{3}{4}$

Here $z^2+z+1-a=0$

$\Rightarrow z=\frac{-1\pm \sqrt{4a-3}}{2}$

Here $a<\frac{3}{4}$

Hence $a\ne \frac{3}{4}$ otherwise $z$ will be purely real.