Let $m$ and $M$ be respectively the minimum and maximum values of $\left|\begin{array}{ccc}\cos ^{2} \mathrm{x} & 1+\sin ^{2} \mathrm{x} & \sin 2 \mathrm{x} \\ 1+\cos ^{2} \mathrm{x} & \sin ^{2} \mathrm{x} & \sin 2 \mathrm{x} \\ \cos ^{2} \mathrm{x} & \sin ^{2} \mathrm{x} & 1+\sin 2 \mathrm{x}\end{array}\right|$. Then the ordered pair $(\mathrm{m}, \mathrm{M})$ is equal to:

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Determinants

Let

m

and

M

be respectively the minimum and maximum values of

cos^{2} x 1 + cos^{2} x cos^{2} x 1 + sin^{2} x sin^{2} x sin^{2} x sin 2 x sin 2 x 1 + sin 2 x

. Then the ordered pair

(m, M)

is equal to:

Answer: (B)

Simplify with the row operations

R_{1} \to R_{1} - R_{3}

and

R_{2} \to R_{2} - R_{3}

, which turn the first two rows into

(0, 1, - 1)

and

(1, 0, - 1)

. Expanding the reduced determinant gives

- 2 - sin 2 x

. Since

sin 2 x

ranges over

[- 1, 1]

, the value

- 2 - sin 2 x

ranges over

[- 3, - 1]

, so the minimum is

m = - 3

and the maximum is

M = - 1

, giving

(m, M) = (- 3, - 1)

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Single correct option