Question

Consider the system of linear equation in x, y and z

(sin 3 θ)x – y + z = 0

(cos 2 θ) x + 4y + 3z = 0

2x + 7y + 7z = 0

If the system has non-trivial solution, then θ ∈ [0, π] are –

• A. 0, π, π/6
• B. 0, π, π/6, 5π/6
• C. 0, π/6, 5π/6
• D. None

Answer 1

Answer: (B)

0, π, π/6, 5π/6

Answer 2

The system has a non-trivial solution is

∆ = $\left| \begin{matrix} \sin 3\theta & –1 & 1 \\ \cos 2\theta & 4 & 3 \\ 2 & 7 & 7 \end{matrix} \right|$ = 0

⇒ 7 sin 3θ + 7 cos 2θ – 6 + 7 cos 2θ – 8 = 0

⇒ sin 3θ + 2 cos 2θ = 2 ⇒ 3 sin θ – 4 sin³θ + 2 (1 – 2sin²θ)

= 2

⇒ sin θ (– 4 sin²θ – 4 sin θ + 3) = 0 ⇒ sin θ = 0,

$\frac{1}{2}$ sin θ ≠ – $\frac{3}{2}$

Int [0, π], θ = 0, π, π/6, 5π/6