1
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1 $${\tan ^{ - 1}}\left( {{{1 + \sqrt 3 } \over {3 + \sqrt 3 }}} \right) + {\sec ^{ - 1}}\left( {\sqrt {{{8 + 4\sqrt 3 } \over {6 + 3\sqrt 3 }}} } \right)$$ is equal to :

A
$${\pi \over 2}$$
B
$${\pi \over 3}$$
C
$${\pi \over 6}$$
D
$${\pi \over 4}$$
2
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1 The sum of the absolute maximum and absolute minimum values of the function $$f(x)=\tan ^{-1}(\sin x-\cos x)$$ in the interval $$[0, \pi]$$ is :

A
0
B
$$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\frac{\pi}{4}$$
C
$$\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)-\frac{\pi}{4}$$
D
$$\frac{-\pi}{12}$$
3
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1 Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $$\cos ^{-1}(x)-2 \sin ^{-1}(x)=\cos ^{-1}(2 x)$$ is equal to :

A
0
B
1
C
$$\frac{1}{2}$$
D
$$-\frac{1}{2}$$
4
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1 If $$0 < x < {1 \over {\sqrt 2 }}$$ and $${{{{\sin }^{ - 1}}x} \over \alpha } = {{{{\cos }^{ - 1}}x} \over \beta }$$, then the value of $$\sin \left( {{{2\pi \alpha } \over {\alpha + \beta }}} \right)$$ is

A
$$4 \sqrt{\left(1-x^{2}\right)}\left(1-2 x^{2}\right)$$
B
$$4 x \sqrt{\left(1-x^{2}\right)}\left(1-2 x^{2}\right)$$
C
$$2 x \sqrt{\left(1-x^{2}\right)}\left(1-4 x^{2}\right)$$
D
$$4 \sqrt{\left(1-x^{2}\right)}\left(1-4 x^{2}\right)$$
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