Let λ ∈ R . The system of linear equations
2x₁ - 4x₂ + λx₃ = 1
x₁ - 6x₂ + x₃ = 2
λx₁ - 10x₂ + 4x₃ = 3
is inconsistent for:

Solution

Maths sir · 3 steps

Find where Δ = 0, then test each such λ: only a nonzero Δᵢ there means no solution.

1. The rule. If Δ ≠ 0, the system has a unique solution. If Δ = 0 and at least one of Δ₁, Δ₂, Δ₃ is nonzero, it is inconsistent. So first find the values of λ that make Δ = 0.

   Δ = |2, -4, λ; 1, -6, 1; λ, -10, 4| = -28 + 4(4-λ) + λ(6λ - 10) = 2(3λ + 2)(λ - 3)

2. Test the roots. Δ = 0 only at λ = 3 and λ = -2/3. Replacing the first column with (1, 2, 3) gives Δ₁ = 6 - 2λ. Check this at each root.

   λ	Δ	Δ₁ = 6 - 2λ	System
   3	0	0 (also Δ₂ = Δ₃ = 0)	equation 1 + equation 2 = equation 3, so infinitely many solutions
   -2/3	0	22/3 ≠ 0	inconsistent
   any other value	≠ 0	not needed	unique solution

3. So. The system has no solution only for λ = -2/3. That is exactly one negative value, option (b).

The idea to keep. A zero Δ only narrows down the suspects. Each root has to be checked separately: a nonzero Δᵢ means no solution, while dependent equations mean infinitely many.

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