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JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Complex Numbers

If a and b are real numbers such that

(2 + α)^{4} = a + b α

, where

α = \frac{- 1 + i 3}{2}

, then

a + b

is equal to:

(A)9✓ Correct
(B)33
(C)57
(D)24

Answer: (A)

Solution

Here

α = \frac{- 1 + i 3}{2}

is a primitive cube root of unity, and

2 + α = \frac{3}{2} + i \frac{3}{2} = 3 e^{iπ /6}

. Raising to the fourth power,

(2 + α)^{4} = (3)^{4} e^{i 2 π /3} = 9 e^{i 2 π /3} = 9 (- \frac{1}{2} + i \frac{3}{2}) = 9 α

. Comparing with

a + b α

gives

a = 0

and

b = 9

, so

a + b = 9

.

JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Single correct option

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