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JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Complex Numbers

Let

α

and

β

be two roots of the equation

x^{2} + 2 x + 2 = 0

, then

α^{15} + β^{15}

is equal to:

(A)-256✓ Correct
(B)512
(C)-512
(D)256

Answer: (A)

Solution

The roots of

x^{2} + 2 x + 2 = 0

are

- 1 \pm i = 2 e^{\pm i 3 π /4}

. By De Moivre,

α^{15} + β^{15} = (2)^{15} \cdot 2 cos \frac{45 π}{4}

. Reducing the angle,

\frac{45 π}{4} - 10 π = \frac{5 π}{4}

, and

cos \frac{5 π}{4} = - \frac{1}{2}

, so the value is

2^{15/2} \cdot 2 \cdot (- \frac{1}{2}) = - 2^{8} = - 256

.

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Single correct option

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