Question

If the minimum and the maximum values of the function $f:\left[\frac{\pi }{4},\frac{\pi }{2}\right]\to R,$ defined by :
$f\left(\theta \right)=\left|\begin{array}{ccc}-si{n}^2\theta & -1-si{n}^2\theta & 1\\ -co{s}^2\theta & -1-co{s}^2\theta & 1\\ 12& 10& -2\end{array}\right|$
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)

$C_3\to C_3-\left(C_1-C_2\right)$
$f\left(\theta \right)=\left|\begin{array}{ccc}-si{n}^2\theta & -1-si{n}^2\theta & 0\\ -co{s}^2\theta & -1-co{s}^2\theta & 0\\ 12& 10& -4\end{array}\right|$
$=-4\left[\left(1+{\cos }^2\theta \right){\sin }^2\theta -{\cos }^2\theta \left(1+{\sin }^2\theta \right)\right]$
$=-4\left[{\sin }^2\theta +{\sin }^2\theta {\cos }^2\theta -{\cos }^2\theta -{\cos }^2\theta {\sin }^2\theta \right]$
$f(\theta )=4\cos 2\theta$
$\theta \in \left[\frac{\pi }{4},\frac{\pi }{2}\right]$
$2\theta \in \left[\frac{\pi }{2},\pi \right]$
$f(\theta )\in [-4,0]$
$(m,M)=(-4,0)$