Complex Numbers
Sum and Product of Roots
nta_pyq_2024_apr
Grade 11
Question:
Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$, $z\in\mathbb{C}$. Then $4(\alpha^2+\beta^2)$ is equal to:
Step-by-Step Solution
Key Concept: Let $z=x+iy$. $(\bar{z})^2+|z|=0\Rightarrow x^2-y^2-2ixy+\sqrt{x^2+y^2}=0$. Cases: $x=0$ gives $-y^2+|y|=0\Rightarrow y=0,\pm1$; $y=0$ gives $x^2+|x|=0\Rightarrow x=0$.
Non-zero roots: $i,-i$. $\alpha=0$, $\beta=1$. $4(\alpha^2+\beta^2)=4$.
Correct Answer: 4