Let for some real numbers α and β, a = α - iβ. If the system of equations 4ix + (1 + i)y = 0 and 8( cos 2π/3 + i sin 2π/3 )x + a̅ y = 0 has more than one solution, then α/β is equal to

Solution

Maths sir · 4 steps

Both equations have zero on the right, so more than one solution means the determinant must vanish.

1. Use the determinant. A homogeneous system always has x = y = 0 as a solution. It has another one only when the coefficient determinant is zero.

   |4i, 1+i; 8(-1/2+√3/2i), a̅| = 0 ⇒ 4i a̅ = (1+i)(-4+4√3 i)

2. Expand both sides. Since a̅ = α + iβ, the left side is 4i(α + iβ) = -4β + 4α i. The right side is (-4 - 4√3) + (4√3 - 4)i.
3. Compare parts. Real parts give β = 1 + √3. Imaginary parts give α = √3 - 1.
4. So. α/β = (√3-1)/(√3+1) = (√3-1)²/2 = (4 - 2√3)/2 = 2 - √3, which is option (b).

The idea to keep. For a homogeneous system, more than one solution means the determinant is zero. With complex coefficients, set the determinant to zero, then equate real and imaginary parts.

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