If the system of linear equations x + 2y + 3z = 4 2x + 3y + 4z = 5 3x + 4y + az = b has infinitely many solutions, and is a complex root of x^2 + x + 1 = 0 , then for real numbers p and q satisfying p ^ a+1 + q ^ b+1 + 2 ^ a+b = 0 , the value of p^2 + q^2 + p + q is equal to

Step-by-step solution

For the given system of equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero ( = 0 ). = | array ccc 1 & 2 & 3 2 & 3 & 4 3 & 4 & a array | = 0 Expanding along the first row: 1(3a - 16) - 2(2a - 12) + 3(8 - 9) = 0 3a - 16 - 4a + 24 - 3 = 0 -a + 5 = 0 a = 5 For infinitely many solutions, we also require _z = 0 : _z = | array ccc 1 & 2 & 4 2 & 3 & 5 3 & 4 & b array | = 0 1(3b - 20) - 2(2b - 15) + 4(8 - 9) = 0 3b - 20 - 4b + 30 - 4 = 0 -b + 6 = 0 b = 6 Now, is a complex roo