The number of real values λ, such that the system of linear equations<br>2x - 3y + 5z = 9<br>x + 3y - z = -18<br>3x - y + (λ<sup>2</sup> - |λ|)z = 16<br>has no solution, is :-
Step-by-Step Solution
Key Concept: A system of linear equations has no solution if the determinant of the coefficient matrix is zero and the system is inconsistent (i.e., the augmented matrix has a row of the form [0 0 0 | k] where k is non-zero).
The coefficient matrix is A = [[2, -3, 5], [1, 3, -1], [3, -1, λ<sup>2</sup> - |λ|]]. The determinant |A| = 2(3(λ<sup>2</sup> - |λ|) - 1) + 3(1(λ<sup>2</sup> - |λ|) + 3) + 5(-1 - 9) = 6λ<sup>2</sup> - 6|λ| - 2 + 3λ<sup>2</sup> - 3|λ| + 9 - 50 = 9λ<sup>2</sup> - 9|λ| - 43. Setting |A| = 0 gives 9λ<sup>2</sup> - 9|λ| - 43 = 0. Let t = |λ| ≥ 0, then 9t<sup>2</sup> - 9t - 43 = 0. The roots are t = (9 ± √(81 - 4(9)(-43))) / 18 = (9 ± √(81 + 1548)) / 18 = (9 ± √1629) / 18. Since √1629 > 9, one root is positive and one is negative. Since t = |λ| ≥ 0, only the positive root is valid, giving two values for λ (±t). Checking consistency for these values shows the system is inconsistent.
Correct Answer: (3)