Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
Let $a, b, c$ be distinct complex numbers with $|a| = |b| = |c| = 1$ and $z_1, z_2$ be the roots of the equation $az^2 + bz + c = 0$ with $|z_1| = 1$. Let $P$ and $Q$ represent the complex numbers $z_1$ and $z_2$ in the Argand plane with $\angle POQ = 0$, $0° < \theta < 180°$ (where $O$ being the origin). Then
$b^2 = ac; \theta = \frac{2\pi}{3}$
$\theta = \frac{2\pi}{3}; PQ = \sqrt{3}$
$PQ = 2\sqrt{3}; b^2 = ac$
$\theta = \frac{\pi}{3}; b^2 = ac$
Step-by-Step Solution
Key Concept: Analyzing binomial expansions modulo powers of 2 determines parity constraints on exponents.
We analyze $3^p = (4-1)^p = 4x_1 + (-1)^p$, $5^q = (4+1)^q = 4x_2 + 1$, and $7^r = (8-1)^r = 8x_3 + (-1)^r$. From the second equation, $5^q \equiv 1 \pmod{4}$ for all $q$, so it's always even. For the first and third, both $p$ and $r$ must be odd or both even. Therefore $p+r$ is always even, while $p+q+r$ can be either odd or even.
Correct Answer: 1,2