Question

Consider the system of equations in x, y, z as
$xsin3\theta -y+z=0,xcos2\theta +4y+3z=0$,
$2x+7y+7z=0.$ If this system has a non-trivial solution, then for integer n, values of $\theta$ are given by

• A. $\pi \left(n+\frac{{\left(-1\right)}^n}{3}\right)$
• B. $\pi \left(n+\frac{{\left(-1\right)}^n}{4}\right)$
• C. $\pi \left(n+\frac{{\left(-1\right)}^n}{6}\right)$
• D. $\frac{n\pi }{2}$

Answer 1

Answer: (C)

Given system of equations
$xsin3\theta -y+z=0$, $xcos2\theta +4y+3z=0$, $2x+7y+7z=0\to$ are homogeneous system of linear equation
Since system has non-trivial solution
$\therefore \left|\begin{array}{ccc}sin3\theta & -1& 1\\ cos2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0$
$\Rightarrow sin3\theta \left[28-21\right]+1\left[7cos2\theta -6\right]+\left[7cos2\theta -8\right]=0$
$\Rightarrow 3sin\theta -4si{n}^3\theta +2\left(1-2si{n}^2\theta \right)-2=0$
$\Rightarrow sin\theta \left(4si{n}^2\theta +4sin\theta -3\right)=0$
Either $sin\theta =0$ or $4sin2\theta +6sin\theta -2sin\theta -3=0$
$\Rightarrow \left(2sin\theta -1\right)\left(2sin\theta +3\right)=0$
$\therefore sin\theta =\frac{1}{2},sin\theta \ne -\frac{3}{2}\left[\because sin\theta >-1\right]$
$\therefore \theta =n\pi$ or $\theta =n\pi +{\left(-1\right)}^n\frac{\pi }{6}$
$\Rightarrow \theta =\pi \left[n+\frac{{\left(-1\right)}^n}{6}\right].$