Question

The determinant $\left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ \sin \theta & x& 1\\ \cos \theta & -1& x\end{array}\right|$
is independent of

• A. $x$
• B. $\theta$
• C. $\text{both}x\text{and}\theta$
• D. $\text{None of these}$

Answer 1

The correct option is B $\theta$

$\Delta =\left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ \sin \theta & x& 1\\ \cos \theta & -1& x\end{array}\right|=x(x^2+1)-\sin \theta (x\sin \theta -\cos \theta )+\cos \theta (-\sin \theta -x\cos \theta )=x^3+x-x({\sin }^2\theta +{\cos }^2\theta )=x^3$
Therefore, Determinant is independent of $\theta$