Question

If the minimum and the maximum values of the function $f :\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow R ,$ defined by :
$f \left(\theta\right)=\begin{vmatrix}-sin^{2}\,\theta&-1-sin^{2}\,\theta&1\\ -cos^{2}\,\theta&-1-cos^{2}\,\theta&1\\ 12&10&-2\end{vmatrix}$
are $m$ and $M$ respectively, then the ordered pair $( m , M )$ is equal to :

• A. $(0,4)$
• B. $(-4,4)$
• C. $(0,2 \sqrt{2})$
• D. $(-4,0)$

Answer 1

Answer: (D)

$C _{3} \rightarrow C _{3}-\left( C _{1}- C _{2}\right)$
$f \left(\theta\right)=\begin{vmatrix}-sin^{2}\,\theta&-1-sin^{2}\,\theta&0\\ -cos^{2}\,\theta&-1-cos^{2}\,\theta&0\\ 12&10&-4\end{vmatrix}$
$=-4\left[\left(1+\cos ^{2} \theta\right) \sin ^{2} \theta-\cos ^{2} \theta\left(1+\sin ^{2} \theta\right)\right]$
$=-4\left[\sin ^{2} \theta+\sin ^{2} \theta \cos ^{2} \theta-\cos ^{2} \theta-\cos ^{2} \theta \sin ^{2} \theta\right]$
$f (\theta)=4 \cos 2 \theta$
$\theta \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$
$2 \theta \in\left[\frac{\pi}{2}, \pi\right]$
$f(\theta) \in[-4,0]$
$( m , M )=(-4,0)$