Let m and M be respectively the minimum and

Question:

Let $\mathrm{m}$ and $\mathrm{M}$ be respectively the minimum and maximum values of

$\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right| .$ Then the

ordered pair $(\mathrm{m}, \mathrm{M})$ is equal to

  1. $(-3,-1)$

  2. $(-4,-1)$

  3. $(1,3)$

  4. $(-3,3)$


Correct Option:

Solution:

$\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$

$\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}, \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$

$\left|\begin{array}{ccc}-1 & 1 & 0 \\ 1 & 0 & -1 \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$

$=-1\left(\sin ^{2} x\right)-1\left(1+\sin 2 x+\cos ^{2} x\right)$

$=-\sin 2 x-2$

$\mathrm{m}=-3, \mathrm{M}=-1$

 

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