1
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1 The area of the region given by

$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$ is :

A
$$\frac{31}{8}$$
B
$$\frac{17}{6}$$
C
$$\frac{19}{6}$$
D
$$\frac{27}{8}$$
2
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1 For any real number $$x$$, let $$[x]$$ denote the largest integer less than equal to $$x$$. Let $$f$$ be a real valued function defined on the interval $$[-10,10]$$ by $$f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } .\end{array}\right.$$ Then the value of $$\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x \,d x$$ is :

A
4
B
2
C
1
D
0
3
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1 The slope of the tangent to a curve $$C: y=y(x)$$ at any point $$(x, y)$$ on it is $$\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 x}}$$. If $$C$$ passes through the points $$\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\right)$$ and $$\left(\alpha, \frac{1}{2} \mathrm{e}^{2 \alpha}\right)$$, then $$\mathrm{e}^{\alpha}$$ is equal to :

A
$$\frac{3+\sqrt{2}}{3-\sqrt{2}}$$
B
$$\frac{3}{\sqrt{2}}\left(\frac{3+\sqrt{2}}{3-\sqrt{2}}\right)$$
C
$$\frac{1}{\sqrt{2}}\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)$$
D
$$\frac{\sqrt{2}+1}{\sqrt{2}-1}$$
4
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1 Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ externally and also touches the $$x$$-axis be $$\mathrm{L}$$. Then the area bounded by $$\mathrm{L}$$ and the line $$y=4$$ is:

A
$$\frac{32 \sqrt{2}}{3}$$
B
$$\frac{40 \sqrt{2}}{3}$$
C
$$\frac{64}{3}$$
D
$$\frac{32}{3}$$
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