Q46 / 160 · AP EAMCET 2018 (24 Apr Shift 1) · Mathematics
A set of values of $\theta$ for which the system of equations $(\sin 3 \theta) x-y+z=0$, $(\cos 2 \theta) x+4 y+3 z=0,2 x+7 y+7 z=0$ has non-trivial solutions, is
- A$(n+1) \frac{\pi}{2}+(-1)^n \frac{\pi}{4}$ (here $n$ is any integer)
- B$(n-1) \frac{\pi}{2}+(-1)^n \frac{\pi}{3}$ (here $n$ is any integer)
- C$\frac{n \pi}{2}+(-1)^n \frac{\pi}{6}$ (here $n$ is any integer)
- Correct answerD$n \pi+(-1)^n \frac{\pi}{6}$ (here $n$ is any integer)
Solution
For non-trivial solution, $\Delta=0$
$$
\left|\begin{array}{ccc}
\sin 3 \theta & -1 & 1 \\
\cos 2 \theta & 4 & 3 \\
2 & 7 & 7
\end{array}\right|=0
$$
On expanding,
$$
\begin{aligned}
& \Rightarrow \sin 3 \theta(28-21)+1(7 \cos 2 \theta-6) \\
& +1(7 \cos 2 \theta-8)=0 \\
& \Rightarrow \quad 7 \sin 3 \theta+14 \cos 2 \theta-14=0 \\
& \Rightarrow \quad \sin 3 \theta+2 \cos 2 \theta-2=0 \\
& \Rightarrow \quad 3 \sin \theta-4 \sin ^3 \theta-2(1-\cos 2 \theta)=0 \\
& \Rightarrow \quad 3 \sin \theta-4 \sin ^3 \theta-4 \sin ^2 \theta=0 \\
& \Rightarrow \quad-\sin \theta\left(4 \sin ^2 \theta+4 \sin \theta-3\right)=0 \\
&
\end{aligned}
$$
$\begin{aligned} & \Rightarrow \sin \theta=0, \quad 4 \sin ^2 \theta+4 \sin \theta-3=0 \\ & \Rightarrow \sin \theta=0, \quad(2 \sin \theta-1)(2 \sin \theta+3)=0 \\ & \Rightarrow \quad \sin \theta=0, \quad \sin \theta=\frac{1}{2}[2 \sin \theta+3 \neq 0] \\ & \Rightarrow \quad \theta=n \pi, \theta=n \pi+(-1)^n \frac{\pi}{6} .\end{aligned}$
Hence, the correct answer is option: 'D'.