1
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $$x$$ satisfying $$\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$$ is :

A
more than 2
B
2
C
0
D
1
2
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
If the domain of the function

$f(x)=\log _{e}\left(4 x^{2}+11 x+6\right)+\sin ^{-1}(4 x+3)+\cos ^{-1}\left(\frac{10 x+6}{3}\right)$ is $(\alpha, \beta]$, then

$36|\alpha+\beta|$ is equal to :
A
72
B
54
C
45
D
63
3
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}$$.

If $$\mathrm{n(S)}$$ denotes the number of elements in $$\mathrm{S}$$ then :

A
$$\mathrm{n}(\mathrm{S})=0$$
B
$$\mathrm{n}(\mathrm{S})=1$$ and only one element in $$\mathrm{S}$$ is less than $$\frac{1}{2}$$.
C
$$\mathrm{n}(\mathrm{S})=1$$ and the elements in $$\mathrm{S}$$ is more than $$\frac{1}{2}$$.
D
$$\mathrm{n}(\mathrm{S})=1$$ and the element in $$\mathrm{S}$$ is less than $$\frac{1}{2}$$.
4
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$S$$ be the set of all solutions of the equation $$\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^{2}}\right)=\pi, x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$$. Then $$\sum_\limits{x \in S} 2 \sin ^{-1}\left(x^{2}-1\right)$$ is equal to :

A
$$\pi-2 \sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$$
B
$$\pi-\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)$$
C
$$\frac{-2 \pi}{3}$$
D
None
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