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 condition for consistency (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 zero: <br> | 2 1 -1 | <br> | 2 -5 \lambda | = 0 <br> | 1 2 -5 | <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, for consistency, the augmented matrix must have rank 2. Using the first and third equations: <br> 2x + y = 5 + z <br> x + 2y = 7 + 5z <br> Solving for x and y: <br> 2(2x + y) - (x + 2y) = 2(5+z) - (7+5z) <br> 3x = 10 + 2z - 7 - 5z = 3 - 3z => x = 1 - z <br> y = 5 + z - 2(1 - z) = 5 + z - 2 + 2z = 3 + 3z <br> Substitute into the second equation: 2(1 - z) - 5(3 + 3z) + 17z = \mu <br> 2 - 2z - 15 - 15z + 17z = \mu <br> -13 = \mu <br> Thus, \lambda = 17, \mu = -13. <br> (\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: A