The sum of all possible values of [0, 2 ] , for which the system of equations : x 3 - 8y - 12z = 0 x 2 + 3y + 3z = 0 x + y + 3z = 0 has a non-trivial solution, is equal to :

Step-by-step solution

For the given system of homogeneous linear equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero. vmatrix 3 & -8 & -12 2 & 3 & 3 1 & 1 & 3 vmatrix = 0 Expanding the determinant along the first row: 3 (9 - 3) - (-8)(3 2 - 3) + (-12)( 2 - 3) = 0 6 3 + 24 2 - 24 - 12 2 + 36 = 0 6 3 + 12 2 + 12 = 0 Dividing by 6 , we get: 3 + 2 2 + 2 = 0 Using the multiple angle formulas 3 = 4 ^3 - 3 and 2 = 2 ^2 - 1 , we substitute these into the equation: (4 ^3 - 3 ) + 2(2 ^2 - 1) + 2 = 0 4