JEE Main 2024 · 31 January · evening · Single correct · +4 / −1

Let z₁ and z₂ be two complex numbers such that z₁+z₂=5 and z₁³+z₂³=20+15 i Then, |z₁⁴+z₂⁴| equals -

  1. (a) 15 √15
  2. (b) 30 √3
  3. (c) 25 √3
  4. (d) 75
Answer and solution

Answer (d) 75

Solution

Maths sir · 4 steps

Do not hunt for z₁ and z₂. Work with their sum and product, and every power sum follows.

  1. Find the product. Let s = z₁ + z₂ = 5 and p = z₁ z₂. The cube identity is z₁³ + z₂³ = s³ - 3ps.

    125 - 15p = 20 + 15i ⇒ 15p = 105 - 15i ⇒ p = 7 - i

  2. Sum of squares. z₁² + z₂² = s² - 2p = 25 - 14 + 2i = 11 + 2i.
  3. Sum of fourth powers. z₁⁴ + z₂⁴ = (z₁² + z₂²)² - 2p². Here (11 + 2i)² = 117 + 44i and p² = (7 - i)² = 48 - 14i, so the result is 117 + 44i - 96 + 28i = 21 + 72i.
  4. So. |21 + 72i| = √(441 + 5184) = √5625 = 75, which is option (d).

The idea to keep. When a question gives symmetric expressions in z₁, z₂, switch to s = z₁ + z₂ and p = z₁ z₂. Every power sum, like z₁ⁿ + z₂ⁿ, can be built from these two.

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 2024.