Question

The set of equation $\lambda x-y+\left(\cos \theta \right)z=0,3x+y+2z=0,\left(\cos \theta \right)x+y+2z=0,0\le \theta <2\pi$, has non-trivial solution(s)

• A. for no value of $\lambda$ and $\theta$
• B. for all values of $\lambda$ and $\theta$
• C. for all values of $\lambda$ and only two values of $\theta$
• D. for only one value of $\lambda$ and all values of $\theta$

Answer 1

The correct option is A for no value of $\lambda$ and $\theta$

Given system of equations $AX=O$
where $A=\left[\begin{array}{ccc}\lambda & -1& \cos \theta \\ 3& 1& 2\\ \cos \theta & 1& 2\end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],O=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
For non-trivial solution,
$D=0$
$\Rightarrow \left|\begin{array}{ccc}\lambda & -1& \cos \theta \\ 3& 1& 2\\ \cos \theta & 1& 2\end{array}\right|=0$
$\Rightarrow {\cos }^2\theta -\cos \theta -6=0$
$\Rightarrow (\cos \theta -3)(\cos \theta +2)=0$
$\Rightarrow \cos \theta =3,-2$
which is not possible.
Hence, there is no value of $\theta$ and $\lambda$ for non-trivial solution.