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Functions question

2022 · Shift 1 · Q32

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PYQ Library
JEE Advanced
Mathematics
Functions
2022 · Shift 1 · Q32

Functions question

2022 · Shift 1 · Q32

JEE AdvancedMathematicsFunctionsMultiple correct+4 / −2

Let

|M|

denote the determinant of a square matrix

M

. Let

g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}

be the function defined by

g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}

where

f(\theta)=\frac{1}{2}\left|\begin{array}{ccc} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{array}\right|+\left|\begin{array}{ccc} \sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi \end{array}\right| .

Let

p(x)

be a quadratic polynomial whose roots are the maximum and minimum values of the function

g(\theta)

, and

p(2)=2-\sqrt{2}

. Then, which of the following is/are TRUE ?

A
$p\left(\frac{3+\sqrt{2}}{4}\right)\lt 0$
B
$p\left(\frac{1+3 \sqrt{2}}{4}\right)\gt 0$
C
$p\left(\frac{5 \sqrt{2}-1}{4}\right)\gt 0$
D
$p\left(\frac{5-\sqrt{2}}{4}\right)\lt 0$

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