If the system of equation <br> 2x + y - z = 5 <br> 2x - 5y + λz = μ <br> x + 2y - 5z = 7 <br> has infinitely many solutions, then (λ + μ)<sup>2</sup> + (λ - μ)<sup>2</sup> is equal to
Step-by-Step Solution
Key Concept: For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the augmented matrix must satisfy the consistency condition (rank(A) = rank(A|B) < number of variables).
The system is: <br> 2x + y - z = 5 <br> 2x - 5y + \lambda z = \mu <br> x + 2y - 5z = 7 <br> For infinitely many solutions, the determinant of the coefficient matrix must be 0: <br> |2 1 -1; 2 -5 \lambda; 1 2 -5| = 0 <br> 2(25 - 2\lambda) - 1(-10 - \lambda) - 1(4 + 5) = 0 <br> 50 - 4\lambda + 10 + \lambda - 9 = 0 <br> 51 - 3\lambda = 0 => \lambda = 17 <br> Now, using the consistency condition, the augmented matrix [A|B] must have rank 2. <br> [2 1 -1 | 5; 2 -5 17 | \mu; 1 2 -5 | 7] <br> R1 -> R1 - 2R3: [0 -3 9 | -9] => -3y + 9z = -9 => y - 3z = 3 <br> R2 -> R2 - 2R3: [0 -9 27 | \mu - 14] <br> For consistency, R2 must be 3 times R1: \mu - 14 = 3(-9) = -27 => \mu = -13 <br> Then (\lambda + \mu)<sup>2</sup> + (\lambda - \mu)<sup>2</sup> = (17 - 13)<sup>2</sup> + (17 + 13)<sup>2</sup> = 4<sup>2</sup> + 30<sup>2</sup> = 16 + 900 = 916.
Correct Answer: 1