If the system of equations
2 x+y-z=5
2 x-5 y+λ z=μ
x+2 y-5 z=7
has infinitely many solutions, then (λ+μ)²+(λ-μ)² is equal to

Solution

Maths sir · 4 steps

Infinite solutions need D = 0 and Dₓ = 0: the first fixes λ, the second fixes μ.

1. Condition. Three equations in three unknowns have infinitely many solutions only when the coefficient determinant D is zero, and Dₓ, D_y and D_z are zero too.

   D = |2, 1, -1; 2, -5, λ; 1, 2, -5| = 2(25 - 2λ) - 1(-10 - λ) - 1(4 + 5) = 51 - 3λ

2. Find lambda. 51 - 3λ = 0 gives λ = 17.
3. Find mu. Replace the x-column with the constants 5, μ, 7 and put λ = 17. Then Dₓ = 5(25 - 34) - 1(-5μ - 119) - 1(2μ + 35) = 39 + 3μ, and Dₓ = 0 gives μ = -13. As a check, D_z = -39 - 3μ is also zero when μ = -13.
4. So. (λ+μ)² + (λ-μ)² = 2(λ² + μ²) = 2(289 + 169) = 916, which is option (a).

The idea to keep. For infinitely many solutions, D = 0 alone is not enough. D = 0 fixes the coefficient, and Dₓ = 0 (or D_y, D_z) fixes the constant on the right side.

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