Question

If $f\left(x\right)=\left|\begin{array}{ccc}{\cos }^{2}x& \cos x.\sin x& -\sin x\\ \cos x.\sin x& {\sin }^{2}x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$, then for all $xϵR$, the value of $f\left(x\right)=$

• A. $0$
• B. $1$
• C. $-1$
• D. None of the above

Answer 1

Answer: (B)

$1$

We have $f\left(x\right)=\left|\begin{array}{ccc}{\cos }^{2}x& \sin x\cos x& -\sin x\\ \cos x\sin x& {\sin }^{2}x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$

Applying $C_1\to C_1-\sin x{C}_3$ and $C_2\to C_2+\cos x.C_3$

$=\left|\begin{array}{ccc}1& 0& -\sin x\\ 0& 1& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$

Again applying $R_3\to R_3-\sin x.R_1$

$=\left|\begin{array}{ccc}1& 0& -\sin x\\ 0& 1& \cos x\\ 0& -\cos x& {\sin }^{2}x\end{array}\right|$

Expanding along $C_1$

$={\sin }^{2}x+{\cos }^{2}x=1$ for all $x$