Let θ ∈ ( 0,π/2 ). If the system of linear equations
(1 + cos ²θ )x + sin ²θ y + 4 sin 3 θ z = 0
cos ²θ x + (1 + sin ²θ )y + 4 sin 3 θ z = 0
cos ²θ x + sin ²θ y + (1 + 4 sin 3 θ )z = 0
has a non-trivial solution, then the value of θ is :

Solution

Maths sir · 5 steps

Add all three columns into the first: every entry becomes 2 + 4 sin 3θ, and the determinant falls out.

1. Condition to use. A homogeneous system has a non-trivial solution only when its coefficient determinant is zero. So set Δ = 0.
2. Column operation. Apply C₁ → C₁ + C₂ + C₃. Each row of the new first column becomes 1 + cos² θ + sin² θ + 4 sin 3θ = 2 + 4 sin 3θ. Take it out as a common factor.

   Δ = (2 + 4 sin 3θ)|1, sin² θ, 4 sin 3θ; 1, 1+sin² θ, 4 sin 3θ; 1, sin² θ, 1+4 sin 3θ|

3. Row operations. Do R₂ → R₂ - R₁ and R₃ → R₃ - R₁. Rows 2 and 3 become (0, 1, 0) and (0, 0, 1), so the remaining determinant is 1 and Δ = 2 + 4 sin 3θ.
4. Solve. 2 + 4 sin 3θ = 0 gives sin 3θ = -1/2. Since θ ∈ (0, π/2), 3θ ∈ (0, 3π/2), and the only angle there with sine -1/2 is 3θ = 7π/6.
5. So. θ = 7π/18, which is option (b).

The idea to keep. When every row has the same entries except for an extra 1 on the diagonal, add all columns into one. The determinant becomes 1 plus the sum of those entries.

Answer from the NTA final answer key. Solution by PrepWiser.