JEE Main 2022 · 29 July · morning · Single correct · +4 / −1

If z=2+3 i, then z⁵+(z̄)⁵ is equal to :

  1. (a) 244
  2. (b) 224
  3. (c) 245
  4. (d) 265
Answer and solution

Answer (a) 244

Solution

Maths sir · 3 steps

Adding z⁵ to its conjugate cancels the imaginary part, so you only need twice the real part of z⁵.

  1. Pair the conjugates. The conjugate of a power is the power of the conjugate, so (z̄)⁵ = z̅⁵̅. A number plus its conjugate is twice its real part, so z⁵ + (z̄)⁵ = 2 Re(z⁵).
  2. Square twice. z² = (2+3i)² = 4 - 9 + 12i = -5 + 12i. Then z⁴ = (-5+12i)² = 25 - 144 - 120i = -119 - 120i.

    z⁵ = (-119 - 120i)(2 + 3i) = -238 - 357i - 240i + 360 = 122 - 597i

  3. So. Re(z⁵) = 122, so z⁵ + (z̄)⁵ = 2 × 122 = 244. The answer is (a).

The idea to keep. zⁿ + (z̄)ⁿ is always 2 Re(zⁿ), a real number. When the angle is not a standard one, build the power by repeated squaring and track only the real part.

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 2022.