Complex Numbers
De Moivre's Theorem – Modulus-Argument Conditions
Complex Numbers_PYQ
Grade 11
Question:
If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \dfrac{\pi}{2}$, then
$\bar{z}w = -i$
$z\bar{w} = \dfrac{1-i}{\sqrt{2}}$
$\bar{z}w = i$
$z\bar{w} = \dfrac{-1+i}{\sqrt{2}}$
Step-by-Step Solution
Key Concept: Modulus of $\bar{z}w$ equals $|zw|=1$, and the argument is $-\arg(z)+\arg(w)=-\pi/2$. Together these uniquely identify the value as $-i$.
**Step 1: Find modulus of z̄w**
$|\bar{z}w| = |z||w| = |zw| = 1$.
**Step 2: Find argument of z̄w**
$\arg(\bar{z}w) = -\arg(z) + \arg(w) = -(\arg(w)+\tfrac{\pi}{2}) + \arg(w) = -\dfrac{\pi}{2}$.
**Step 3: Conclude**
$\bar{z}w$ has modulus $1$ and argument $-\pi/2$, so $\bar{z}w = e^{-i\pi/2} = -i$.
Correct Answer: 1