Question

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

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

Answer 1

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

The system of equations has non-trivial solutions iff
$\left|\begin{array}{ccc}\lambda & -1& \cos \theta \\ 3& 1& 2\\ \cos \theta & 1& 2\end{array}\right|=0$

$\Rightarrow 6-2\cos \theta +\cos \theta (3-\cos \theta )=0$
$\Rightarrow \cos \theta -{\cos }^2\theta +6=0$
LHS $>0$ for all $\theta$ as $|\cos \theta |\le 1$
So, the system of equations cannot have non-trivial solutions for any value of $\lambda$ and $\theta$