1
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

Let $$x=\frac{m}{n}$$ ($$m, n$$ are co-prime natural numbers) be a solution of the equation $$\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}$$ and let $$\alpha, \beta(\alpha >\beta)$$ be the roots of the equation $$m x^2-n x-m+ n=0$$. Then the point $$(\alpha, \beta)$$ lies on the line

A
$$3 x-2 y=-2$$
B
$$3 x+2 y=2$$
C
$$5 x+8 y=9$$
D
$$5 x-8 y=-9$$
2
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
3
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
4
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 $$\frac{1}{2}$$.
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