1
JEE Main 2022 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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
2
JEE Main 2022 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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
$$\frac{\sqrt{2}+1}{\sqrt{2}-1}$$
3
JEE Main 2022 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The general solution of the differential equation $$\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0$$ is :

A
$$\left(y^{2}+x\right)^{4}=\mathrm{C}\left|\left(y^{2}+2 x\right)^{3}\right|$$
B
$$\left(y^{2}+2 x\right)^{4}=C\left|\left(y^{2}+x\right)^{3}\right|$$
C
$$\left|\left(y^{2}+x\right)^{3}\right|=\mathrm{C}\left(2 y^{2}+x\right)^{4}$$
D
$$\left|\left(y^{2}+2 x\right)^{3}\right|=C\left(2 y^{2}+x\right)^{4}$$
4
JEE Main 2022 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

A line, with the slope greater than one, passes through the point $$A(4,3)$$ and intersects the line $$x-y-2=0$$ at the point B. If the length of the line segment $$A B$$ is $$\frac{\sqrt{29}}{3}$$, then $$B$$ also lies on the line :

A
$$2 x+y=9$$
B
$$3 x-2 y=7$$
C
$$ x+2 y=6$$
D
$$2 x-3 y=3$$
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