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 from the orthocentre of the triangle formed by the lines $x=0$, $y=0$ and $x+y=1$ is

1

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)

2

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)

3

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)

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

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)

6

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)

7

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)

8

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)

9

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)

10

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)

1

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)

2

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)