If the minimum and the maximum values of the function $f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$, defined by $f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|$ are $m$ and $M$ respectively, then the ordered pair $(m, M)$ is equal to:

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Determinants

If the minimum and the maximum values of the function

f : [\frac{π}{4}, \frac{π}{2}] \to R

, defined by

f (θ) = - sin^{2} θ - cos^{2} θ 12 - 1 - sin^{2} θ - 1 - cos^{2} θ 10 11 - 2

are

m

and

M

respectively, then the ordered pair

(m, M)

is equal to:

Answer: (D)

Add the first column to the third; using

1 - sin^{2} θ = cos^{2} θ

and

1 - cos^{2} θ = sin^{2} θ

and simplifying, the determinant collapses to

f (θ) = 4 cos 2 θ

. On

[\frac{π}{4}, \frac{π}{2}]

the angle

2 θ

runs over

[\frac{π}{2}, π]

, where

cos 2 θ

falls from

0

- 1

. Hence

f

ranges over

[- 4, 0]

, so the minimum and maximum are

m = - 4

and

M = 0

, giving the pair

(- 4, 0)

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option