JEE Main 2022 · 29 June · evening · Single correct · +4 / −1

Let α be a root of the equation 1 + x² + x⁴ = 0. Then, the value of α¹⁰¹¹ + α²⁰²² - α³⁰³³ is equal to :

  1. (a) 1
  2. (b) α
  3. (c) 1 + α
  4. (d) 1 + 2α
Answer and solution

Answer (a) 1

Solution

Maths sir · 3 steps

Multiply the equation by x² - 1 to get α⁶ = 1, then reduce every power modulo 6.

  1. Find a period. Multiply 1 + x² + x⁴ by x² - 1. The product is x⁶ - 1, and the second factor is zero at x = α.

    (α² - 1)(α⁴ + α² + 1) = α⁶ - 1 = 0 ⇒ α⁶ = 1

  2. Reduce the powers. 1011 = 6 × 168 + 3, so α¹⁰¹¹ = α³. 2022 = 6 × 337, so α²⁰²² = 1. 3033 = 6 × 505 + 3, so α³⁰³³ = α³.
  3. So. The expression is α³ + 1 - α³ = 1. The α³ terms cancel, so you never need to know whether α³ is 1 or -1. The answer is (a).

The idea to keep. For 1 + x² + x⁴ = 0 or 1 + x + x² = 0, multiply by x² - 1 or x - 1 to get x⁶ = 1 or x³ = 1. Then cut big exponents down to their remainders.

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.