If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + λz = μ
has more than two solutions, then μ - λ² is equal to ______.

Solution

Maths sir · 5 steps

A linear system never has exactly two solutions, so more than two means infinitely many.

1. What it means. A linear system has no solution, exactly one, or infinitely many. More than two solutions forces infinitely many, so we need D = 0 and D₃ = 0 (and also D₁ = D₂ = 0).

   D = |1, 1, 1; 1, 2, 3; 3, 2, λ| = (2λ - 6) - (λ - 9) + (2 - 6) = λ - 1

2. Find lambda. D = 0 gives λ = 1.
3. Find mu. Replace the z-column by the constants: D₃ = |1, 1, 6; 1, 2, 10; 3, 2, μ| = (2μ - 20) - (μ - 30) + 6(2 - 6) = μ - 14. Setting D₃ = 0 gives μ = 14. With these values D₁ and D₂ also come out as 0.
4. Quick check. The third equation should be a combination of the first two. Taking 4 × (first) - (second) gives 3x + 2y + z = 14, which matches λ = 1 and μ = 14.
5. So. μ - λ² = 14 - 1 = 13.

The idea to keep. For infinitely many solutions, D = 0 is not enough. The constants must also make D₁, D₂ and D₃ zero, or the system has no solution.

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