Question:

Let $z$ be a complex number with nonzero imaginary part and $a=z^2+z+1$ is real. Then '$a$' cannot take the value

$-1$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$\dfrac{3}{4}$

Step-by-Step Solution

Key Concept: $\text{Im}(z^2+z)=0$ with $\text{Im}(z)\neq0$. Writing $z=x+iy$: $y(2x+1)=0 \Rightarrow x=-1/2$.

$a<3/4$; cannot take $a=3/4$.

Correct Answer: 4