The set of equations λx−y+(cos⁡θ)z=0,3x+y+2z=0,(cos⁡θ)x+y+2z=0 ; 0≤θ<2π has non-trivial solution(s)

The determinant of the coefficient matrix is D=λ−1cos⁡θ312cos⁡θ12=λ(2−2)+1(6−2cos⁡θ)+cos⁡θ(3−cos⁡θ)=6−cos2⁡θ+cos⁡θ=6+14−cos2⁡θ−cos⁡θ+14Converting into a perfect square, we get=254−cos⁡θ−122 Minimum value of D is 254−−1−122=4>0i.eD≠0∴ Given system has only trivial solutions  For no values of λ and θ system has non trivial solution(s)  Therefore, the correct answer is (1).