Question

$Q.If\left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ -\sin \theta & -x& 1\\ \cos \theta & 1& x\end{array}\right|=8,writethevalueofx.$

Answer 1

$$\begin{gathered} \left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ -\sin \theta & -x& 1\\ \cos \theta & 1& x\end{array}\right|=8 \\ Expandalong{R}_1 \\ x\left(-x^2-1\right)-\sin \theta \left(-x\sin \theta -\cos \theta \right)+\cos \theta \left(-\sin \theta +x\cos \theta \right)=8 \\ -x^3-x+x{\sin }^2\theta +\sin \theta \cos \theta -\sin \theta \cos \theta +x{\cos }^2\theta =8 \\ -x^3-x+x\left({\sin }^2\theta +{\cos }^2\theta \right)=8 \\ -x^3-x+x=8 \\ -x^3=8 \\ {x}^3=-8 \\ x=-2 \end{gathered}$$