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 (i.e., the augmented matrix rank is not equal to the coefficient matrix rank).
The system is: <br> x + y + z = 1 <br> x + ay + z = 1 <br> ax + by + z = 0 <br> The determinant of the coefficient matrix 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 = 2a - a^2 - 1 = -(a-1)^2. <br> For no solution, D = 0, so a = 1. <br> If a = 1, the equations become: <br> x + y + z = 1 <br> x + y + z = 1 <br> x + by + z = 0 <br> Subtracting the first from the third gives (b-1)y = -1. <br> For no solution, we need 0*y = constant (non-zero), which happens if b-1 = 0, i.e., b = 1. <br> If b = 1, we get 0 = -1, which is impossible. Thus, S = {1}, which is a singleton set.
Correct Answer: 1