Let $f:[-\tfrac{\pi}{2}, \tfrac{\pi}{2}] \to \mathbb{R}$ be a differentiable function such that $f(0)=\tfrac{1}{2}$. If $\displaystyle \lim_{x \to 0} \frac{x \int_0^x f(t),dt}{e^{x^2} - 1} = \alpha$, then $8\alpha^2$ is equal to:

1

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)

2

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)

3

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)

4

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)

5

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)

6

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)

7

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)

8

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)

9

Let $g:\mathbb{R}\to\mathbb{R}$ be a non-constant twice-differentiable function such that $g'\!\left(\tfrac12\right)=g'…

Topic: JEE Main 2024 (30 January Morning Shift)

1

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

Topic: JEE Main 2024 (30 January Morning Shift)

2

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity …

Topic: JEE Main 2024 (30 January Morning Shift)

3

Two integers $x$ and $y$ are chosen with replacement from the set ${0,1,2,3,\ldots,10}$. Then the probability that $|x-…

Topic: JEE Main 2024 (30 January Morning Shift)

4

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$. If the diagonal $\overrightarrow{BD}= \…

Topic: JEE Main 2024 (30 January Morning Shift)

5

Let $y=y(x)$ be the solution of the differential equation $\sec x,dy+{2(1-x)\tan x+x(2-x)},dx=0$ with $y(0)=2$. Then $y…

Topic: JEE Main 2024 (30 January Morning Shift)

6

If $z=x+iy$ with $xy\ne0$ satisfies $z^{2}+i\overline{z}=0$, then $|z^{2}|$ equals:

Topic: JEE Main 2024 (30 January Morning Shift)

7

If $2\sin^3x+\sin2x\cos x+4\sin x-4=0$ has exactly $3$ solutions in the interval $\left[0,\dfrac{n\pi}{2}\right],,n\in\…

Topic: JEE Main 2024 (30 January Morning Shift)

8

If the domain of the function $f(x)=\cos^{-1}!\left(\dfrac{2-|x|}{4}\right)+{\log_e(3-x)}^{-1}$ is $[-\alpha,\beta)-{\g…

Topic: JEE Main 2024 (30 January Morning Shift)

9

Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15…

Topic: JEE Main 2024 (30 January Morning Shift)

10

The area (in square units) of the region bounded by the parabola $y^{2}=4(x-2)$ and the line $y=2x-8$ is:

Topic: JEE Main 2024 (30 January Morning Shift)