Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x\in\mathbb{R}$. If $\lim_{n\to\infty}{f(x)-f\left(\dfrac{x}{2^{n}}\right)}=G(x)$, then $\displaystyle \sum_{r=1}^{10} G(r^{2})$ is equal to

1

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)

2

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)

3

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)

4

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)

5

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)

6

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

Topic: JEE Main 2025 (4 April Morning Shift)

7

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)

8

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)

9

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)…

Topic: JEE Main 2025 (4 April Morning Shift)

10

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)