JEE Main 2023 · 31 January · evening · Single correct · +4 / −1

The complex number z=(i-1)/(cos π/3+i sin π/3) is equal to :

  1. (a) cos π/12-i sin π/12
  2. (b) √2(cos π/12+i sin π/12)
  3. (c) √2 i(cos (5 π)/12-i sin (5 π)/12)
  4. (d) √2(cos (5 π)/12+i sin (5 π)/12)
Answer and solution

Answer (d) √2(cos (5 π)/12+i sin (5 π)/12)

Solution

Maths sir · 3 steps

Put the top in polar form, then dividing just means dividing moduli and subtracting arguments.

  1. Numerator in polar form. i - 1 = -1 + i has modulus √(1+1) = √2. It lies in the second quadrant, so its argument is π - π/4 = 3π/4. That gives i - 1 = √2 e^(i 3π/4).
  2. Denominator in polar form. cos π/3 + i sin π/3 = e^(iπ/3), with modulus 1 and argument π/3.

    z = (√2 e^(i 3π/4))/e^(iπ/3) = √2 e^(i(3π/4 - π/3)) = √2 e^(i((9π - 4π)/12)) = √2 e^(i 5π/12)

  3. So. z = √2(cos 5π/12 + i sin 5π/12), which is option (d).

The idea to keep. In polar form, multiplying adds arguments and dividing subtracts them. Always find the quadrant before you write the argument: -1 + i is at 3π/4, not π/4.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Complex Numbers

Asked about 160 times in JEE Main since 2020. This question is from 2023.