QJEEM20Ma3359: JEE Main 2020 Mathematics - Class 12 JEEM20Ma3359 Solution: Let m and M be respectively the minimum and maximum values of

Question

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Let m and M be respectively the minimum and maximum values of

| \begin{matrix} \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{matrix} |

Then the ordered pair (m, M) is equal to :

Accepted Answer

| \begin{matrix} \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{matrix} |

R₁

\to

R₁ – R₂, R₂

\to

R₂ – R₃

| \begin{matrix} - 1 & 1 & 0 \\ 1 & 0 & - 1 \\ \cos^{2} x & \sin^{2} x & 1 + \sin 2 x \end{matrix} |

= –1(sin² x) – 1(1 + sin2x + cos² x)

= - sin2x - 2

∴

minimum value when sin2x = 1

m = - 2 - 1 = -3

∴

Maximum value when sin2x = –1

M = -2 + 1 = -1

∴

(m, M) = (–3, –1)

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