JEE Main 2025 Jan 22 Shift 1, Mathematics Q4: Introduction to Complex Numbers And Their Conjugates

JEE Main2025Jan 22, Shift 1Mathematics

Let $z_{1}, z_{2}$ and $z_{3}$ be three complex numbers on the circle $∣ z ∣ = 1$ with $ar g (z_{1}) = - \frac{π}{4}$ , $ar g (z_{2}) = 0$ and $ar g (z_{3}) = \frac{π}{4}$ . If $∣ z_{1} \overline{z_{2}} + z_{2} \overline{z_{3}} + z_{3} \overline{z_{1}} ∣^{2} = α + β 2$ , $α, β \in Z$ , then the value of $α^{2} + β^{2}$ is :

A
24
B
41
C
31
D
29

Solution

Writing the three unit complex numbers as $z_{1} = e^{- iπ /4}$ , $z_{2} = 1$ , $z_{3} = e^{iπ /4}$ .

$z_{1} \overline{z_{2}} = e^{- iπ /4}$ , $z_{2} \overline{z_{3}} = e^{- iπ /4}$ , $z_{3} \overline{z_{1}} = e^{iπ /4} \cdot e^{iπ /4} = e^{iπ /2} = i$ .

Sum $= 2 e^{- iπ /4} + i = 2 (\frac{1}{2} - \frac{i}{2}) + i = 2 + (1 - 2) i$ .

Modulus squared $= (2)^{2} + (1 - 2)^{2} = 2 + 1 - 22 + 2 = 5 - 22$ .

So $α = 5$ , $β = - 2$ , and $α^{2} + β^{2} = 25 + 4 = 29$ .

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JEE Main 2025 Jan 22 Shift 1, Mathematics Q4: Introduction to Complex Numbers And Their Conjugates

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