Question

$\left|\begin{array}{ccc}0& \sin x-\cos x& \sin x+\cos x\\ {\sin }^{2}x& \sin x& \cos x\\ {\cos }^{2}x& -\cos x& \sin x\end{array}\right|$ is equal to

• A. $\sin x$
• B. $1$
• C. $-1$
• D. $\cos x+\sin x$

Answer 1

The correct option is C $-1$

$\left|\begin{array}{ccc}0& \sin x-\cos x& \sin x+\cos x\\ {\sin }^{2}x& \sin x& \cos x\\ {\cos }^{2}x& -\cos x& \sin x\end{array}\right|$
Using row operations,
$R_1\to R_1-(R_2+R_3)$
$=\left|\begin{array}{ccc}-1& 0& 0\\ {\sin }^{2}x& \sin x& \cos x\\ {\cos }^{2}x& -\cos x& \sin x\end{array}\right|=-1({\sin }^{2}x+{\cos }^{2}x)=-1$