1
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1

The domain of the function $$f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$$, where [t] is the greatest integer function, is :

A
$$\left(-\sqrt{\frac{5}{2}}, \frac{5-\sqrt{5}}{2}\right)$$
B
$$\left(\frac{5-\sqrt{5}}{2}, \frac{5+\sqrt{5}}{2}\right)$$
C
$$\left(1, \frac{5-\sqrt{5}}{2}\right)$$
D
$$\left[1, \frac{5+\sqrt{5}}{2}\right)$$
2
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
$$4 \sqrt{\left(1-x^{2}\right)}\left(1-4 x^{2}\right)$$
3
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

$$\tan \left(2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{\sqrt{5}}{2}+2 \tan ^{-1} \frac{1}{8}\right)$$ is equal to :

A
1
B
2
C
$$\frac{1}{4}$$
D
$$\frac{5}{4}$$
4
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

Let m and M respectively be the minimum and the maximum values of $$f(x) = {\sin ^{ - 1}}2x + \sin 2x + {\cos ^{ - 1}}2x + \cos 2x,\,x \in \left[ {0,{\pi \over 8}} \right]$$. Then m + M is equal to :

A
$$1 + \sqrt 2 + \pi$$
B
$$\left( {1 + \sqrt 2 } \right)\pi$$
C
$$\pi + \sqrt 2$$
D
$$1 + \pi$$
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