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 $\Delta =0$
$\begin{vmatrix} sin \alpha & -cos ⁡ \alpha & 3 \\ cos ⁡ \alpha & sin ⁡ \alpha & -2 \\ 2 & 3 & cos ⁡ \alpha \end{vmatrix}=0$
$\Rightarrow sin \alpha \left(sin ⁡ \alpha cos ⁡ \alpha + 6\right)+cos ⁡ \alpha \left(\left(cos\right)^{2} ⁡ \alpha + 4\right)+3\left(3 cos ⁡ \alpha - 2 sin ⁡ \alpha \right)=0$
$\Rightarrow sin^{2} \alpha cos ⁡ \alpha +6sin ⁡ \alpha +cos^{3} ⁡ \alpha +4cos ⁡ \alpha +9cos ⁡ \alpha -6sin ⁡ \alpha =0$
$\Rightarrow 14cos \alpha =0\Rightarrow cos⁡\alpha =0\Rightarrow \alpha =\left(2 n + 1\right)\frac{\pi }{2},n\in I$
$\Rightarrow $ Number of values of $\alpha $ in $\left[- 10 \pi , 10 \pi \right]$ is $20$