Let $f:(-\infty,\infty)\setminus{0}\to\mathbb{R}$ be differentiable such that $f'(1)=\lim_{a\to\infty} a^{2}f!\left(\tfrac{1}{a}\right)$. Then $\displaystyle \lim_{a\to\infty}\left(\frac{a(a+1)}{2}\tan^{-1}!\frac{1}{a}+a^{2}-2\log_{e}a\right)$ is:

1

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\al…

Topic: JEE Main 2024 (6 April Morning Shift)

2

Let $\alpha,\beta$ be the distinct roots of $x^{2}-(t^{2}-5t+6)x+1=0$, $t\in\mathbb{R}$, and let $a_n=\alpha^{n}+\beta^…

Topic: JEE Main 2024 (6 April Morning Shift)

3

Let $y=y(x)$ solve $(2x\log_e x),\dfrac{dy}{dx}+2y=\dfrac{3}{x}\log_e x$ for $x>0$ with $y(e^{-1})=0$. Then $y(e)$ equa…

Topic: JEE Main 2024 (6 April Morning Shift)

4

The shortest distance between the lines $\dfrac{x-3}{2}=\dfrac{y+15}{-7}=\dfrac{z-9}{5}$ and $\dfrac{x+1}{2}=\dfrac{y-1…

Topic: JEE Main 2024 (6 April Morning Shift)

5

$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\righ…

Topic: JEE Main 2024 (6 April Morning Shift)

6

Let the area of the region enclosed by the curves $y=3x$, $2y=27-3x$ and $y=3x-x\sqrt{x}$ be $A$. Then $10A$ is:

Topic: JEE Main 2024 (6 April Morning Shift)

7

Let the relations $R_1$ and $R_2$ on the set $X={1,2,3,\ldots,20}$ be given by $R_1={(x,y):,2x-3y=2}$ and $R_2={(x,y):,…

Topic: JEE Main 2024 (6 April Morning Shift)

8

If $A(3,1,-1),\ B!\left(\tfrac{5}{3},\tfrac{7}{3},\tfrac{1}{3}\right),\ C(2,2,1)$ and $D!\left(\tfrac{10}{3},\tfrac{2}{…

Topic: JEE Main 2024 (6 April Morning Shift)

9

Let $C$ be the circle of minimum area touching the parabola $y=6-x^{2}$ and the lines $y=\sqrt{3},|x|$. Which of the fo…

Topic: JEE Main 2024 (6 April Morning Shift)

10

Let $A={,n\in[100,700]\cap\mathbb N:\ n\text{ is neither a multiple of }3\text{ nor a multiple of }4,}$. Then the numbe…

Topic: JEE Main 2024 (6 April Morning Shift)

1

$\displaystyle \int_{0}^{\pi/4}\frac{\cos^{2}x,\sin^{2}x}{\big(\cos^{3}x+\sin^{3}x\big)^{2}},dx$ is equal to:

Topic: JEE Main 2024 (6 April Morning Shift)

2

The mean and standard deviation of $20$ observations are found to be $10$ and $2$, respectively. On rechecking, one obs…

Topic: JEE Main 2024 (6 April Morning Shift)

3

Let $y=y(x)$ solve the differential equation $(1+x^{2})\dfrac{dy}{dx}+y=e^{\tan^{-1}x}$ with $y(1)=0$. Then $y(0)$ is:

Topic: JEE Main 2024 (6 April Morning Shift)

4

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of th…

Topic: JEE Main 2024 (6 April Morning Shift)

5

The function $f(x)=\dfrac{x^{2}+2x-15}{x^{2}-4x+9},\ x\in\mathbb{R}$ is:

Topic: JEE Main 2024 (6 April Morning Shift)

6

The interval in which the function $f(x)=x^{x}$, $x>0$, is strictly increasing is:

Topic: JEE Main 2024 (6 April Morning Shift)

7

A company has two plants $A$ and $B$ to manufacture motorcycles. $60%$ are made at $A$ and $40%$ at $B$. Of these, $80%…

Topic: JEE Main 2024 (6 April Morning Shift)