Let n denote the number of solutions of the equation z² + 3z̅ = 0, where z is a complex number. Then the value of Σₖ₌₀^∞ 1/nᵏ is equal to :

Solution

Maths sir · 5 steps

Put z = x + iy, split into real and imaginary parts, and count every solution, including z = 0.

1. Split the equation. With z = x + iy, we get z² = x² - y² + 2ixy and 3z̅ = 3x - 3iy. Both the real part and the imaginary part of the sum must be zero.

   x² - y² + 3x = 0, y(2x - 3) = 0

2. Case y = 0. The real part gives x² + 3x = 0, so x = 0 or x = -3. That gives z = 0 and z = -3.
3. Case x = 3/2. The real part gives 9/4 - y² + 9/2 = 0, so y² = 27/4 and y = ±3√3/2. That gives two more solutions, z = 3/2 ± 3√3/2i.
4. Sum the series. So n = 4. The sum is a geometric series with first term 1 and ratio 1/4, so it equals 1/(1 - 1/4) = 4/3.
5. So. The value is 4/3, which is option (b).

The idea to keep. When an equation mixes z and z̅, put z = x + iy and equate real and imaginary parts. Work through every case, and never forget to check z = 0.

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