Let $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ and let $A(\alpha,\beta),\ B(1,0),\ C(\gamma,\delta),\ D(1,2)$ be the vertices of a parallelogram $ABCD$. If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to:

1

Let $S$ be the set of positive integral values of $a$ for which $\frac{a x^{2}+2(a+1)x+9a+4}{x^{2}-8x+32}

Topic: JEE Main 2024 (31 January Morning Shift)

2

For $\alpha,\beta,\gamma\ne 0$, if $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)\,(\a…

Topic: JEE Main 2024 (31 January Morning Shift)

3

$\displaystyle \lim_{x\to 0}\frac{e^{\,2|\sin x|}-2|\sin x|-1}{x^{2}}$

Topic: JEE Main 2024 (31 January Morning Shift)

1

For $0 < c < b < a$, let $(a+b-2c)x^{2} + (b+c-2a)x + (c+a-2b) = 0$ and let $\alpha \ne 1$ be one of its roots. Then, a…

Topic: JEE Main 2024 (31 January Morning Shift)

2

If $f(x)=\dfrac{4x+3}{6x-4}$, $x\ne\dfrac{2}{3}$, and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\left\{\dfrac{2}{3}\rig…

Topic: JEE Main 2024 (31 January Morning Shift)

3

Let $y=y(x)$ be the solution of the differential equation $\displaystyle \frac{dy}{dx}=\frac{\tan x + y}{\sin x}$, $x\i…

Topic: JEE Main 2024 (31 January Morning Shift)

4

Three rotten apples are accidentally mixed with fifteen good apples. Assuming the random variable $x$ to be the number …

Topic: JEE Main 2024 (31 January Morning Shift)

5

Let $\vec a=3\hat i+\hat j-2\hat k,\ \vec b=4\hat i+\hat j+7\hat k$ and $\vec c=\hat i-3\hat j+4\hat k$ be three vector…

Topic: JEE Main 2024 (31 January Morning Shift)

6

The sum of the series $\displaystyle \frac{1}{1-3\cdot1^{2}+1^{4}}+\frac{2}{1-3\cdot2^{2}+2^{4}}+\frac{3}{1-3\cdot3^{2}…

Topic: JEE Main 2024 (31 January Morning Shift)

7

Two marbles are drawn in succession from a box containing $10$ red, $30$ white, $20$ blue and $15$ orange marbles, with…

Topic: JEE Main 2024 (31 January Morning Shift)

8

The distance of the point $Q(0,2,-2)$ from the line passing through the point $P(5,-4,3)$ and perpendicular to the line…

Topic: JEE Main 2024 (31 January Morning Shift)

9

Let $g(x)$ be a linear function and $ f(x)= \begin{cases} g(x), & x\le 0,\\[2mm] \left(\dfrac{1+x}{2+x}\right)^{\tfrac{…

Topic: JEE Main 2024 (31 January Morning Shift)

10

The area of the region $\Big\{(x,y): y^{2}\le4x,\ x0,\ x\ne3\Big\}$ is:

Topic: JEE Main 2024 (31 January Morning Shift)