Let S={z ∈ ℂ: z²+z̄=0}. Then Σ_(z ∈ S)(Re(z)+Im(z)) is equal to ______________.

Solution

Maths sir · 3 steps

Put z = x + iy, split into real and imaginary parts, and factor the imaginary equation.

1. Split the equation. With z = x + iy, z² = x² - y² + 2ixy and z̄ = x - iy. A complex number is zero only when its real part and its imaginary part are both zero.

   x² - y² + x = 0, y(2x - 1) = 0

2. Two cases. If y = 0, then x² + x = 0, so x = 0 or x = -1. If x = 1/2, then 1/4 - y² + 1/2 = 0, so y² = 3/4 and y = ±√3/2.

   z	Re(z)	Im(z)	Re(z) + Im(z)
   0	0	0	0
   -1	-1	0	-1
   1/2 + √3/2i	1/2	√3/2	1/2 + √3/2
   1/2 - √3/2i	1/2	-√3/2	1/2 - √3/2

3. So. The √3 terms cancel, leaving 0 - 1 + 1/2 + 1/2 = 0. The sum is 0.

The common slip: 1. Cancelling y from 2xy = y throws away the y = 0 roots, including z = -1, and gives 1 instead of 0.

The idea to keep. When an equation mixes z and z̄, write z = x + iy and equate both parts to zero. Factor, never divide by a variable, or you lose roots.

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