Question

If (√3​−i)ⁿ=2ⁿ, then n is a multiple of:
(a) 2
(b) 3
(c) 4
(d) 5

Answer 1

To determine the value of n, we can equate the real and imaginary parts of both sides of the equation. By setting the imaginary part to 0, we find that n = 0 and the answer is (d) 5.

Step-by-step explanation:

To determine the value of n, we can equate the real and imaginary parts of both sides of the equation. Since √3 is a positive real number, the real part of (√3 - i)ⁿ is 2ⁿ and the imaginary part is 0. We can write:

Real part: (√3)ⁿ * cos(nθ) = 2ⁿ

Imaginary part: -1 * (√3)ⁿ * sin(nθ) = 0

The imaginary part will be 0 if sin(nθ) = 0, which is true when nθ is a multiple of π. Therefore, nθ = kπ, where k is an integer.

Since we're given that (√3 - i)ⁿ = 2ⁿ, nθ = 0, and θ = 0, we can substitute these values into the equation:

n(0) = 0

n = 0

So, n = 0 and the answer is (d) 5.