If the system of equation, a<sup>2</sup>x - ay = 1 - a & bx + (3 - 2b)y = 3 + a possess a unique solution x = 1, y = 1 than :
Step-by-Step Solution
Key Concept: Substitute x=1 and y=1 into both equations to find the values of a and b, then verify the uniqueness condition (determinant of the coefficient matrix is non-zero).
Substituting x=1, y=1 into the equations: <br> 1) a<sup>2</sup>(1) - a(1) = 1 - a => a<sup>2</sup> - a = 1 - a => a<sup>2</sup> = 1 => a = 1 or a = -1. <br> 2) b(1) + (3 - 2b)(1) = 3 + a => b + 3 - 2b = 3 + a => 3 - b = 3 + a => -b = a => b = -a. <br> If a = 1, then b = -1. If a = -1, then b = 1. <br> Check uniqueness: The coefficient matrix is [[a^2, -a], [b, 3-2b]]. Determinant = a^2(3-2b) - (-a)(b) = 3a^2 - 2a^2b + ab. <br> For a=1, b=-1: Det = 3(1)^2 - 2(1)^2(-1) + (1)(-1) = 3 + 2 - 1 = 4 != 0. (Unique) <br> For a=-1, b=1: Det = 3(-1)^2 - 2(-1)^2(1) + (-1)(1) = 3 - 2 - 1 = 0. (Not unique) <br> Thus, a=1, b=-1 is the only solution.
Correct Answer: 1