Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function such that $f(x)=1-2x+\displaystyle\int_{0}^{x}e^{,x-t}f(t),dt$ for all $x\in[0,\infty)$. Then the area of the region bounded by $y=f(x)$ and the coordinate axes is

1

The value of $\displaystyle \int_{-1}^{1}\frac{(1+\sqrt{|x|}-x)e^{x}+(\sqrt{|x|}-x)e^{-x}}{e^{x}+e^{-x}},dx$ is equal to

Topic: JEE Main 2025 (4 April Morning Shift)

2

If $10\sin^4\theta+15\cos^4\theta=6$, then the value of $\dfrac{27\csc^6\theta+8\sec^6\theta}{16\sec^8\theta}$ is

Topic: JEE Main 2025 (4 April Morning Shift)

3

$1+3+5^2+7+9^2+\cdots$ upto $40$ terms is equal to

Topic: JEE Main 2025 (4 April Morning Shift)

4

Consider the sets $A={(x,y)\in\mathbb{R}\times\mathbb{R}:x^{2}+y^{2}=25}$, $B={(x,y)\in\mathbb{R}\times\mathbb{R}:x^{2}…

Topic: JEE Main 2025 (4 April Morning Shift)

5

Consider the equation $x^{2}+4x-n=0$, where $n\in[20,100]$ is a natural number. Then the number of all distinct values …

Topic: JEE Main 2025 (4 April Morning Shift)

6

The length of the latus–rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\dfrac{4}{5}$, …

Topic: JEE Main 2025 (4 April Morning Shift)

7

Let $A={1,6,11,16,\ldots}$ and $B={9,16,23,30,\ldots}$ be the sets consisting of the first $2025$ terms of two arithmet…

Topic: JEE Main 2025 (4 April Morning Shift)

8

Let the shortest distance between the lines $\dfrac{x-3}{3}=\dfrac{y-\alpha}{-1}=\dfrac{z-3}{1}$ and $\dfrac{x+3}{-3}=\…

Topic: JEE Main 2025 (4 April Morning Shift)

9

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x\in\mathbb{R}…

Topic: JEE Main 2025 (4 April Morning Shift)

1

The probability of forming a $12$-person committee from $4$ engineers, $2$ doctors, and $10$ professors containing at l…

Topic: JEE Main 2025 (4 April Morning Shift)

2

A box contains $10$ pens of which $3$ are defective. A sample of $2$ pens is drawn at random and let $X$ denote the num…

Topic: JEE Main 2025 (4 April Morning Shift)

3

Consider two vectors $\vec{u}=3\hat{i}-\hat{j}$ and $\vec{v}=2\hat{i}+\hat{j}-\lambda\hat{k},\ \lambda>0$. The angle be…

Topic: JEE Main 2025 (4 April Morning Shift)

4

For an integer $n\ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2n-3}$ is $16$…

Topic: JEE Main 2025 (4 April Morning Shift)

5

Let $A$ and $B$ be two distinct points on the line $L:\ \dfrac{x-6}{3}=\dfrac{y-7}{2}=\dfrac{z-7}{-2}$. Both $A$ and $B…

Topic: JEE Main 2025 (4 April Morning Shift)

6

Let $f,g:(1,\infty)\to\mathbb{R}$ be defined as $f(x)=\dfrac{2x+3}{5x+2}$ and $g(x)=\dfrac{2-3x}{1-x}$. If the range of…

Topic: JEE Main 2025 (4 April Morning Shift)

7

If $\displaystyle \lim_{x\to 1^{+}}\frac{(x-1)\big(6+\lambda\cos(x-1)\big)+\mu\sin(1-x)}{(x-1)^{3}}=-1$, where $\lambda…

Topic: JEE Main 2025 (4 April Morning Shift)

8

Let the three sides of a triangle be on the lines 4x-7y+10=0, x+y=5 and 7x+4y=15. Then the distance of its orthocentre …

Topic: JEE Main 2025 (4 April Morning Shift)

9

In the expansion of $\left(\sqrt[3]{2}+\dfrac{1}{\sqrt[3]{3}}\right)^{n},\ n\in\mathbb{N}$, if the ratio of $15^{\text{…

Topic: JEE Main 2025 (4 April Morning Shift)

10

Considering the principal values of the inverse trigonometric functions, $\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}…

Topic: JEE Main 2025 (4 April Morning Shift)