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यदि $f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|$ द्वारा परिभाषित फलन $f :\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow R$ के निम्नतम तथा उच्चतम मान क्रमशः $m$ तथा $M$ हैं, तो क्रमित युग्म $( m , M )$ बराबर है
$(0,4)$
$(-4,4)$
$(0,2 \sqrt{2})$
$(-4,0)$
Solution
$C _{3} \rightarrow C _{3}-\left( C _{1}- C _{2}\right)$
$f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 0 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 0 \\ 12 & 10 & -4\end{array}\right|$
$=-4\left[\left(1+\cos ^{2} \theta\right) \sin ^{2} \theta-\cos ^{2} \theta\left(1+\sin ^{2} \theta\right)\right]$
$=-4\left[\sin ^{2} \theta+\sin ^{2} \theta \cos ^{2} \theta-\cos ^{2} \theta-\cos ^{2} \theta \sin ^{2} \theta\right]$
$f(\theta)=4 \cos 2 \theta$
$\theta \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$
$2 \theta \in\left[\frac{\pi}{2}, \pi\right]$
$f(\theta) \in[-4,0]$
$( m , M )=(-4,0)$
