Let the complex number z = x + iy be such that (2z - 3i)/(2z + i) is purely imaginary. If x + y² = 0, then y⁴ + y² - y is equal to :

Solution

Maths sir · 4 steps

Purely imaginary means the real part is zero, so set it to zero and swap x for -y².

1. Write it out. With z = x + iy, the top is 2x + i(2y-3) and the bottom is 2x + i(2y+1). Multiply both by the conjugate of the bottom, 2x - i(2y+1), so the denominator becomes real.
2. Real part. The new numerator has real part 4x² + (2y-3)(2y+1) and imaginary part -8x. For a purely imaginary value the real part must vanish, and x ≠ 0 keeps the imaginary part alive.

   4x² + 4y² - 4y - 3 = 0

3. Use the condition. From x + y² = 0 we get x = -y², so x² = y⁴. The equation becomes 4y⁴ + 4y² - 4y - 3 = 0, that is 4(y⁴ + y² - y) = 3.
4. So. y⁴ + y² - y = 3/4, which is option (c).

The idea to keep. For a quotient of complex numbers, multiply by the conjugate of the denominator. Purely imaginary means real part zero, and purely real means imaginary part zero.

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