Let S be the set of all values of θ ∈[-π, π] for which the system of linear equations
x+y+√3 z=0
-x+(tan θ) y+√7 z=0
x+y+(tan θ) z=0
has non-trivial solution. Then 120/π Σθ ∈ s θ is equal to :

Solution

Maths sir · 4 steps

Non-trivial solution means the determinant is zero. Factor it, then collect every angle in [-π, π].

1. Set determinant zero. A homogeneous system has a non-trivial solution only when its coefficient determinant is 0. Write t = tan θ and subtract row 1 from row 3, so row 3 becomes (0, 0, t-√3).

   |1, 1, √3; -1, t, √7; 0, 0, t-√3| = (t-√3)(t+1) = 0

2. Find every angle. tan θ = √3 gives θ = π/3, -2π/3. tan θ = -1 gives θ = -π/4, 3π/4. Each value of tan repeats every π, so it has two angles in [-π, π].
3. Add them. π/3 - 2π/3 - π/4 + 3π/4 = -π/3 + π/2 = π/6.
4. So. 120/π × π/6 = 20, which is option (d).

The idea to keep. Homogeneous system with a non-trivial solution means determinant = 0. When the answer is a trig equation on an interval, list every solution in that interval, not only the principal value.

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