Let α and β be the roots of the equation x² + (2i - 1) = 0. Then, the value of |α⁸ + β⁸| is equal to :

Solution

Maths sir · 4 steps

There is no x term, so α² = β² = 1 - 2i. Work with x² and the eighth powers come out together.

1. Find the squares. The equation is x² = 1 - 2i, so its two roots are α and β = -α. Both have the same square: α² = β² = 1 - 2i.
2. Eighth powers. Since α⁸ = (α²)⁴, we get α⁸ = β⁸ = (1 - 2i)⁴. So α⁸ + β⁸ = 2(1 - 2i)⁴.

   |α⁸ + β⁸| = 2|1 - 2i|⁴ = 2(√5)⁴ = 2 × 25 = 50

3. Check by expanding. (1 - 2i)² = -3 - 4i and (-3 - 4i)² = -7 + 24i. So α⁸ + β⁸ = -14 + 48i, and √(196 + 2304) = √2500 = 50.
4. So. |α⁸ + β⁸| = 50, which is option (a).

The idea to keep. When roots come as ±√w, every even power of them equals the same power of w. Then use |zⁿ| = |z|ⁿ and you never have to find the roots.

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