Complex Numbers
Range of real expression in z
MMTS_Full_Test_14
Grade 12
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
(A) $-1$
(B) $\dfrac{1}{3}$
(C) $\dfrac{1}{2}$
(D) $\dfrac{3}{4}$
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: (D) $\dfrac{3}{4}$