Let the solution curve $y = y(x)$ of the differential equation $\left(1 + e^{2x}\right)\left(\dfrac{dy}{dx} + y\right) = 1$ pass through the point $\left(0, \dfrac{\pi}{2}\right)$. Then, $\lim_{x \to \infty} e^x y(x)$ is equal to :

1

The integral $\int_{0}^{\tfrac{\pi}{2}} \dfrac{1}{3 + 2 \sin x + \cos x} , dx$ is equal to :

Topic: JEE Main 2022 (29 July Morning Shift)

2

If $\lim_{x \to 0} \dfrac{\alpha e^{x^2} + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \dfrac{2}{3}$, where $\alpha, \b…

Topic: JEE Main 2022 (29 July Morning Shift)

3

If $\dfrac{1}{(20-a)(40-a)} + \dfrac{1}{(40-a)(60-a)} + \cdots + \dfrac{1}{(180-a)(200-a)} = \dfrac{1}{256}$, then the …

Topic: JEE Main 2022 (29 July Morning Shift)

4

Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

Topic: JEE Main 2022 (29 July Morning Shift)

5

If $z = 2 + 3i$, then $z^5 + (\bar{z})^5$ is equal to:

Topic: JEE Main 2022 (29 July Morning Shift)

6

Let $R$ be a relation from the set ${1,2,3,\dots,60}$ to itself such that R={(a,b):b=pq,    where p,q≥3 are prime numbe…

Topic: JEE Main 2022 (29 July Morning Shift)

1

$ \text{Let the solution curve } y=y(x) \text{ of the differential equation } (1+e^{2x})!\left(\dfrac{dy}{dx}+y\right)=…

Topic: JEE Main 2022 (29 July Morning Shift)

2

Let a line L pass through the point of intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0, \mathrm{~b} \in \mathbf…

Topic: JEE Main 2022 (29 July Morning Shift)

3

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP interse…

Topic: JEE Main 2022 (29 July Morning Shift)

4

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{4}$. If $\theta$ is the…

Topic: JEE Main 2022 (29 July Morning Shift)

5

If $f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$ then $f\left(\…

Topic: JEE Main 2022 (29 July Morning Shift)

6

The area of the region$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$ is equal to :

Topic: JEE Main 2022 (29 July Morning Shift)

7

The area of the region$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$ is equal to :

Topic: JEE Main 2022 (29 July Morning Shift)

8

$ \text{Let the focal chord of the parabola } P: y^{2}=4x \text{ along the line } L: y=mx+c,; m>0 \text{ meet the parab…

Topic: JEE Main 2022 (29 July Morning Shift)

9

$ \text{The number of points where the function } f:\mathbb{R}\to\mathbb{R},\quad f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3),|x…

Topic: JEE Main 2022 (29 July Morning Shift)

10

$ \text{The number of points where the function } f:\mathbb{R}\to\mathbb{R},\quad f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3),|x…

Topic: JEE Main 2022 (29 July Morning Shift)