JEE Main 2026 Apr 6 Shift 2, Mathematics Q3: De Moivre's Theorem And Its Applications

JEE Main2026Apr 6, Shift 2Mathematics

Let $S = {z \in C : z^{2} + 6 i z - 3 = 0}$ . Then $z \in S \sum z^{8}$ is equal to:

Solution

Apply the quadratic formula: $z = \frac{- 6 i \pm - 6 + 12}{2} = \frac{- 6 i \pm 6}{2} = \frac{6}{2} (\pm 1 - i)$ .

So $α = \frac{6}{2} (1 - i)$ and $β = - \frac{6}{2} (1 + i)$ .

Express in polar form: $∣ α ∣ = ∣ β ∣ = 3$ , with $α = 3 e^{- iπ /4}$ and $β = 3 e^{i 3 π /4}$ (equivalently $- 3 e^{- iπ /4}$ rotated by $π$ ).

$α^{8} = 3^{4} e^{- i 2 π} = 81$ and $β^{8} = 3^{4} e^{i 6 π} = 81$ .

Sum $= 81 + 81 = 162$ .

Concept behind this question

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