Question

$\begin{vmatrix}\sin^{2}x&\cos^{2}x&1\\ \cos^{2}x&\sin^{2}x&1\\ -10&12&2\end{vmatrix} $ 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 = \begin{vmatrix}\sin^{2}x&\cos^{2}x&1\\ \cos^{2}x&\sin^{2}x&1\\ -10&12&2\end{vmatrix} $
Applying $C_{1} \to C_{1} +C_{2}, $ we get
$ A =\begin{vmatrix}\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{vmatrix}$
$ = \begin{vmatrix}1&\cos^{2}x&1\\ 1&\sin^{2}x&1\\ 2&12&2\end{vmatrix} $
Since, two columns are identical
$\therefore \ A = 0 $