Let M and m respectively be the maximum and the minimum values of… - Vidyadip

MCQ

Let $M$ and $m$ respectively be the maximum and the minimum values of
$f(x)=\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & 1+\cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & \cos ^{2} x & 1+4 \sin 4 x\end{array}\right|, x \in R$
Then $M^{4}-m^{4}$ is equal to :

✓
1280
B
1295
C
1040
D
1215

✓

Answer

Correct option: A.

1280

(A)
Sol. $\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & 1+\cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & \cos ^{2} x & 1+4 \sin 4 x\end{array}\right|, x \in R$
$\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1} \& \mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}$
$f(x)\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 4 x \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right|$
Expand about $\mathrm{R}_{1}$, we get
$f(x)=2+4 \sin 4 x$
$\therefore M=$ max value of $f(x)=6$
$m=\min$ value of $f(x)=-2$
$\therefore \mathrm{M}^{4}-\mathrm{m}^{4}=1280$

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