1
JEE Main 2022 (Online) 29th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $$[t]$$ denotes the greatest integer $$\leq t$$, then the value of $$\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x$$ is :

A
$$\frac{\sqrt{37}+\sqrt{13}-4}{6}$$
B
$$\frac{\sqrt{37}-\sqrt{13}-4}{6}$$
C
$$\frac{-\sqrt{37}-\sqrt{13}+4}{6}$$
D
$$\frac{-\sqrt{37}+\sqrt{13}+4}{6}$$
2
JEE Main 2022 (Online) 29th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

For $$I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$$, if $$I\left(\frac{\pi}{4}\right)=2^{1011}$$, then

A
$$3^{1010} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0$$
B
$$3^{1010} I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0$$
C
$$3^{1011} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0$$
D
$$3^{1011} I\left(\frac{\pi}{6}\right)-I\left(\frac{\pi}{3}\right)=0$$
3
JEE Main 2022 (Online) 29th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If the solution curve of the differential equation $$\frac{d y}{d x}=\frac{x+y-2}{x-y}$$ passes through the points $$(2,1)$$ and $$(\mathrm{k}+1,2), \mathrm{k}>0$$, then

A
$$2 \tan ^{-1}\left(\frac{1}{k}\right)=\log _{e}\left(k^{2}+1\right)$$
B
$$\tan ^{-1}\left(\frac{1}{k}\right)=\log _{e}\left(k^{2}+1\right)$$
C
$$2 \tan ^{-1}\left(\frac{1}{k+1}\right)=\log _{e}\left(k^{2}+2 k+2\right)$$
D
$$2 \tan ^{-1}\left(\frac{1}{k}\right)=\log _{e}\left(\frac{k^{2}+1}{k^{2}}\right)$$
4
JEE Main 2022 (Online) 29th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$y=y(x)$$ be the solution curve of the differential equation $$ \frac{d y}{d x}+\left(\frac{2 x^{2}+11 x+13}{x^{3}+6 x^{2}+11 x+6}\right) y=\frac{(x+3)}{x+1}, x>-1$$, which passes through the point $$(0,1)$$. Then $$y(1)$$ is equal to :

A
$$\frac{1}{2}$$
B
$$\frac{3}{2}$$
C
$$\frac{5}{2}$$
D
$$\frac{7}{2}$$
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