1
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+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
$$p\left(\frac{3+\sqrt{2}}{4}\right)<0$$
B
$$p\left(\frac{1+3 \sqrt{2}}{4}\right)>0$$
C
$$p\left(\frac{5 \sqrt{2}-1}{4}\right)>0$$
D
$$p\left(\frac{5-\sqrt{2}}{4}\right)<0$$
2
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2

Let $$f(x) = \sin \left( {{\pi \over 6}\sin \left( {{\pi \over 2}\sin x} \right)} \right)$$ for all $$x \in R$$ and g(x) = $${{\pi \over 2}\sin x}$$ for all x$$\in$$R. Let $$(f \circ g)(x)$$ denote f(g(x)) and $$(g \circ f)(x)$$ denote g(f(x)). Then which of the following is/are true?

A
Range of f is $$\left[ { - {1 \over 2},{1 \over 2}} \right]$$.
B
Range of f $$\circ$$ g is $$\left[ { - {1 \over 2},{1 \over 2}} \right]$$.
C
$$\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {g(x)}} = {\pi \over 6}$$.
D
There is an x$$\in$$R such that (g $$\circ$$ f)(x) = 1.
3
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-0
For every pair of continuous function f, g : [0, 1] $$\to$$ R such that max {f(x) : x $$\in$$ [0, 1]} = max {g(x) : x $$\in$$ [0, 1]}. The correct statement(s) is (are)
A
[f(c)]2 + 3f(c) = [g(c)]2 + 3g(c) for some c $$\in$$ [0, 1]
B
[f(c)]2 + f(c) = [g(c)]2 + 3g(c) for some c $$\in$$ [0, 1]
C
[f(c)]2 + 3f(c) = [g(c)]2 + g(c) for some c $$\in$$ [0, 1]
D
[f(c)]2 = [g(c)]2 for some c $$\in$$ [0, 1]
4
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-0
Let $$f:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R$$ be given by $$f(x) = {[\log (\sec x + \tan x)]^3}$$. Then,
A
f(x) is an odd function
B
f(x) is a one-one function
C
f(x) is an onto function
D
f(x) is an even function
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