Consider the system of linear equations x+y+z=5, x+2 y+λ² z=9, x+3 y+λ z=μ, where λ, μ ∈ ℝ. Then, which of the following statement is NOT correct?

Solution

Maths sir · 3 steps

D vanishes at λ = 1 and at λ = -1/2, and that second root breaks (c).

1. When D vanishes. Expand the coefficient determinant along the first row: D = (2λ - 3λ²) - (λ - λ²) + (3 - 2) = -2λ² + λ + 1. A unique solution exists only when D ≠ 0.

   D = |1, 1, 1; 1, 2, λ²; 1, 3, λ| = (1-λ)(2λ+1)

2. Subtract equations. Second minus first gives y + (λ² - 1)z = 4, and third minus second gives y + (λ - λ²)z = μ - 9. At λ = 1 both z-coefficients are 0, and at λ = -1/2 both are -3/4. In each case the left sides match, so the system has infinitely many solutions when μ = 13 and none when μ ≠ 13.

   Option	Check	Statement is
   (a) λ ≠ 1, μ = 13	If λ ≠ -1/2, the solution is unique. If λ = -1/2, there are infinitely many. It is consistent either way.	correct
   (b) λ = 1, μ ≠ 13	y = 4 and y = μ - 9 ≠ 4 contradict each other	correct
   (c) λ ≠ 1, μ ≠ 13	λ = -1/2 is allowed, but then D = 0 and the system is inconsistent	NOT correct
   (d) λ = 1, μ = 13	Both reduced equations become y = 4, so z is free	correct

3. So. Take λ = -1/2 with any μ ≠ 13. The system has no solution at all, so statement (c) is the one that is NOT correct.

The idea to keep. In a parameter question, solve D = 0 completely before trusting any 'unique solution' claim. Each root of D is its own case, and the options are built to catch a missed root.

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