Let $f(x)=\left|\begin{array}{ccc}\sin ^{2} x & -2+\cos ^{2} x & \cos 2 x \\ 2+\sin ^{2} x & \cos ^{2} x & \cos 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\cos 2 x\end{array}\right|, x \in[0, \pi]$. Then the maximum value of $f(x)$ is equal to ____.

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Determinants

Let

f (x) = sin^{2} x 2 + sin^{2} x sin^{2} x - 2 + cos^{2} x cos^{2} x cos^{2} x cos 2 x cos 2 x 1 + cos 2 x, x \in [0, π]

. Then the maximum value of

f (x)

is equal to ____.

Answer: 6

Replacing the second row by

R_{2} - R_{1}

and the third by

R_{3} - R_{1}

leaves the determinant unchanged:

f (x) = sin^{2} x 20 cos^{2} x - 2 22 cos 2 x 01

Expanding along the first row,

f (x) = 2 sin^{2} x - 2 (cos^{2} x - 2) + 4 cos 2 x = 2 (sin^{2} x - cos^{2} x) + 4 + 4 cos 2 x

Since

sin^{2} x - cos^{2} x = - cos 2 x

f (x) = - 2 cos 2 x + 4 + 4 cos 2 x = 2 cos 2 x + 4

[0, π]

the greatest value of

cos 2 x

1

, attained at

x = 0

and

x = π

, so the maximum value of

f

6

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Integer answer