Let $g:\mathbb{R}\to\mathbb{R}$ be a non-constant twice-differentiable function such that $g'\!\left(\tfrac12\right)=g'\!\left(\tfrac32\right)$. If a real-valued function $f$ is defined as $f(x)=\dfrac12\,[\,g(x)+g(2-x)\,]$, then

1

The value of $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^{3}}{(n^{2}+k^{2})(n^{2}+3k^{2})}$ is:

Topic: JEE Main 2024 (30 January Morning Shift)

2

Let $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ and $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$ be two vectors such that $|\vec a…

Topic: JEE Main 2024 (30 January Morning Shift)

3

If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then:

Topic: JEE Main 2024 (30 January Morning Shift)

4

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2x^{2}+54…

Topic: JEE Main 2024 (30 January Morning Shift)

5

If \[ f(x)= \begin{vmatrix} 2\cos^{4}x & 2\sin^{4}x & 3+\sin^{2}2x\\ 3+2\cos^{4}x & 2\sin^{4}x & \sin^{2}2x\\ 2\cos^{4}…

Topic: JEE Main 2024 (30 January Morning Shift)

6

Let $(\alpha,\beta,\gamma)$ be the foot of the perpendicular from the point $(1,2,3)$ on the line \[ \frac{x+3}{5}=\fra…

Topic: JEE Main 2024 (30 January Morning Shift)

7

A line passing through the point \(A(9,0)\) makes an angle of \(30^\circ\) with the positive direction of the \(x\)-axi…

Topic: JEE Main 2024 (30 January Morning Shift)

8

Consider the system of linear equations $x + y + z = 4\mu,\quad x + 2y + 2\lambda z = 10\mu,\quad x + 3y + 4\lambda^2 z…

Topic: JEE Main 2024 (30 January Morning Shift)

9

Let $f:[-\tfrac{\pi}{2}, \tfrac{\pi}{2}] \to \mathbb{R}$ be a differentiable function such that $f(0)=\tfrac{1}{2}$. If…

Topic: JEE Main 2024 (30 January Morning Shift)

10

Let M denote the median of the following frequency distributionThen 20M is equal to :

Topic: JEE Main 2024 (30 January Morning Shift)