Question

The number of values of $\alpha$ in $\left[-10\pi ,10\pi \right]$ for which the equations $\left(sin\alpha \right)x-\left(cos\alpha \right)y+3z=0,\left(cos\alpha \right)x+\left(sin\alpha \right)y-2z=0$ and $2x+3y+\left(cos\alpha \right)z=0$ have nontrivial solution is

• A. $10$
• B. $20$
• C. $40$
• D. $15$

Answer 1

Answer: (B)

For non-trivial solution $\Delta =0$
$\left|\begin{array}{ccc}sin\alpha & -cos\alpha & 3\\ cos\alpha & sin\alpha & -2\\ 2& 3& cos\alpha \end{array}\right|=0$
$\Rightarrow sin\alpha \left(sin\alpha cos\alpha +6\right)+cos\alpha \left({\left(cos\right)}^2\alpha +4\right)+3\left(3cos\alpha -2sin\alpha \right)=0$
$\Rightarrow si{n}^2\alpha cos\alpha +6sin\alpha +co{s}^3\alpha +4cos\alpha +9cos\alpha -6sin\alpha =0$
$\Rightarrow 14cos\alpha =0\Rightarrow cos\alpha =0\Rightarrow \alpha =\left(2n+1\right)\frac{\pi }{2},n\in I$
$\Rightarrow$ Number of values of $\alpha$ in $\left[-10\pi ,10\pi \right]$ is $20$