Question

$\left|\begin{array}{ccc}{\sin }^{2}x& {\cos }^{2}x& 1\\ {\cos }^{2}x& {\sin }^{2}x& 1\\ -10& 12& 2\end{array}\right|$ is equal to:

• A. 0
• B. $12{\cos }^{2}x-10{\sin }^{2}x$
• C. $12{\cos }^{2}x-10{\sin }^{2}x-2$
• D. $10\sin 2x$

Answer 1

Answer: (A)

Let $A=\left|\begin{array}{ccc}{\sin }^{2}x& {\cos }^{2}x& 1\\ {\cos }^{2}x& {\sin }^{2}x& 1\\ -10& 12& 2\end{array}\right|$
Applying $C_1\to C_1+C_2,$ we get
$A=\left|\begin{array}{ccc}{\sin }^{2}x+{\cos }^{2}x& {\cos }^{2}x& 1\\ {\cos }^{2}x+{\sin }^{2}x& {\sin }^{2}x& 1\\ -10+12& 12& 2\end{array}\right|$
$=\left|\begin{array}{ccc}1& {\cos }^{2}x& 1\\ 1& {\sin }^{2}x& 1\\ 2& 12& 2\end{array}\right|$
Since, two columns are identical
$\therefore A=0$