Question

The number of q∈ (0, 4π) for which the system of linear equations
3(sin 3θ) x - y + z = 2
3(cos 2θ) x + 4 y + 3 z = 3
6 x + 7 y + 7 z = 9
has no solution, is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

Answer: (B)

7

Answer 2

Δ=$\begin{vmatrix} 3sin\theta&-1 &1 \\\ 3cos2\theta& 4 &3 \\\ 6&7 &7 \end{vmatrix}$
= 3sin3θ(7) + 1(21cos2θ – 18) + 1(21cos2θ – 24)
Δ = 21sin 3θ + 42cos 2θ – 42
For no solution
sin3θ + 2cos2θ = 2
⇒ sin3θ = 2⋅2sin2θ
⇒ 3sinθ – 4sin3θ = 4sin2θ
⇒ sinθ(3 – 4sinθ – 4sin2θ) = 0
sinθ=0 OR sinθ=12
θ=$\pi$,2$\pi$,3$\pi$,$\frac{\pi}{6}$,$\frac{5\pi}{6}$,$\frac{13\pi}{6}$,$\frac{17\pi}{6}$