Complex Numbers
Purely Imaginary Condition with Constraint
nta_pyq_2023_apr
Grade 11
Question:
Let the complex number $z=x+iy$ be such that $\dfrac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$ is equal to
$\dfrac{2}{3}$
$\dfrac{3}{2}$
$\dfrac{3}{4}$
$\dfrac{4}{3}$
Step-by-Step Solution
Key Concept: Set $\text{Re}\!\left(\frac{2z-3i}{2z+i}\right)=0$ to get $4x^2+4y^2-4y-3=0$, i.e., $x^2+y^2-y-\frac{3}{4}=0$.
$x^2+y^2-y=\frac{3}{4}$. With $x=-y^2$: $y^4+y^2-y=\frac{3}{4}$.
Correct Answer: 3