Question

Let $m$ and $M$ be respectively the minimum and maximum values of $\begin{vmatrix}&cos^{2}x&1+sin^{2}x&sin 2x\\ &1+cos^{2}x&sin^{2}x&sin 2x\\ &cos^{2}x&sin^{2}x&1+sin 2x\end{vmatrix}$. Then the ordered pair $( m , M )$ is equal to

• A. (-3,-1)
• B. (-4,-1)
• C. (1,3)
• D. (-3,3)

Answer 1

Answer: (A)

$\begin{vmatrix}&cos^{2}x&1+sin^{2}x&sin 2x\\ &1+cos^{2}x&sin^{2}x&sin 2x\\ &cos^{2}x&sin^{2}x&1+sin 2x\end{vmatrix}$
$R _{1} \rightarrow R _{1}- R _{2}, R _{2} \rightarrow R _{2}- R _{3}$
$\begin{vmatrix}-1&1&0\\ 1&0&-1\\ cos^{2}x&sin^{2}x&1+sin 2 x\end{vmatrix}$
$=-1\left(\sin ^{2} x\right)-1\left(1+\sin 2 x+\cos ^{2} x\right)$
$=-\sin 2 x-2$
$m=-3, M=-1$