1
JEE Main 2024 (Online) 4th April Morning Shift
+4
-1

If the domain of the function $$\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$$ is $$(\alpha, \beta]$$, then $$3 \alpha+10 \beta$$ is equal to:

A
95
B
100
C
97
D
98
2
JEE Main 2024 (Online) 31st January Evening Shift
+4
-1

If $$a=\sin ^{-1}(\sin (5))$$ and $$b=\cos ^{-1}(\cos (5))$$, then $$a^2+b^2$$ is equal to

A
25
B
$$4 \pi^2+25$$
C
$$8 \pi^2-40 \pi+50$$
D
$$4 \pi^2-20 \pi+50$$
3
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

For $$\alpha, \beta, \gamma \neq 0$$, if $$\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$$ and $$(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$$, then $$\gamma$$ equals

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