The number of values of (0, 3 ) for which the system of linear equations x 3 + y 2 + z = 0 x + y + z = 0 2x + 3y + 4z = 0 has non-trivial solutions is

Step-by-step solution

For a homogeneous system of linear equations to have non-trivial solutions, the determinant of the coefficient matrix must be zero. D = vmatrix 3 & 2 & 1 1 & 1 & 1 2 & 3 & 4 vmatrix = 0 Expanding along the first row: 3 (4 - 3) - 2 (4 - 2) + 1 (3 - 2) = 0 3 - 2 2 + 1 = 0 Using multiple angle identities 3 = 4 ^3 - 3 and 2 = 2 ^2 - 1 : (4 ^3 - 3 ) - 2(2 ^2 - 1) + 1 = 0 4 ^3 - 4 ^2 - 3 + 3 = 0 4 ^2 ( - 1) - 3( - 1) = 0 ( - 1)(4 ^2 - 3) = 0 This gives = 1 or = 3 2 . We need to find the number of solutions in the interva