If S is the set of distinct values of 'b' for which the following system of linear equations <br> x + y + z = 1 <br> x + ay + z = 1 <br> ax + by + z = 0 <br> has no solution, then S is :
Step-by-Step Solution
Key Concept: For a system of linear equations to have no solution, the determinant of the coefficient matrix must be zero, and the system must be inconsistent. The determinant is |1 1 1; 1 a 1; a b 1| = 1(a-b) - 1(1-a) + 1(b-a^2) = a - b - 1 + a + b - a^2 = 2a - a^2 - 1 = -(a-1)^2. For this to be zero, a = 1. Substituting a = 1 into the equations: x + y + z = 1, x + y + z = 1, x + by + z = 0. For no solution, the third equation must contradict the first two. If a = 1, the first two equations are identical. The system becomes x + y + z = 1 and x + by + z = 0. For no solution, the planes must be parallel but not coincident, which happens if the coefficients of x, y, z are proportional but the constant term is not. Here, coefficients are (1, 1, 1) and (1, b, 1). For these to be parallel, b must be 1. If b = 1, the equations are x + y + z = 1 and x + y + z = 0, which is a contradiction (no solution). Thus, S = {1}, which is a singleton set.
The system of equations is: <br> x + y + z = 1 <br> x + ay + z = 1 <br> ax + by + z = 0 <br> The determinant of the coefficient matrix is: <br> D = |1 1 1; 1 a 1; a b 1| = 1(a-b) - 1(1-a) + 1(b-a^2) = a - b - 1 + a + b - a^2 = -(a^2 - 2a + 1) = -(a-1)^2. <br> For the system to have no solution, we must have D = 0, which implies a = 1. <br> Substituting a = 1 into the system: <br> x + y + z = 1 <br> x + y + z = 1 <br> x + by + z = 0 <br> The first two equations are identical. The system reduces to: <br> x + y + z = 1 <br> x + by + z = 0 <br> For this system to have no solution, the planes must be parallel, which requires the coefficients of x, y, z to be proportional. <br> 1/1 = 1/b = 1/1, which implies b = 1. <br> If b = 1, the equations are x + y + z = 1 and x + y + z = 0, which is impossible. <br> Thus, for b = 1, there is no solution. <br> Therefore, S = {1}, which is a singleton set.
Correct Answer: (1)