The number of values of θ∈(0,π) for which the system of linear equations x+3y+7z=0, x+4y+7z=0, (sin3θ)x+(cos2θ)y+2z=0 has a non-trivial solution, is :

|1 3 7|
|1 4 7|=0
|sin3θ cos2θ 2|
(8cos2θ)(7) - 7(-cos2θ - sin3θ) = 0
14cos2θ + 21sin3θ = 0
14cos2θ + 21sin3θ = 0
14(1 - 2sin²θ) + 21(3sinθ - 4sin³θ) = 0
14 - 28sin²θ + 63sinθ - 84sin³θ = 0
-84sin³θ - 28sin²θ + 63sinθ + 14 = 0
7sinθ[-12sin²θ - 4sinθ + 9] + 2 = 0
7sinθ[ -12sin²θ - 4sinθ + 9] = 0
7sinθ[2sinθ + 3](2sinθ - 3) = 0
7sinθ(2sinθ + 3)(2sinθ - 3) = 0
sinθ = 0, sinθ = -3/2, sinθ = 3/2
sinθ = 0
Hence, 2 solution in (0,π)