Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$ and $3 \vec{a}-\lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is :

1

If $\displaystyle \int e^{x}\!\left(\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\frac{\sin^{-1}x}{(1-x^{2})^{3/2}}+\frac{x}{1-x^…

Topic: JEE Main 2025 (22 January Evening Shift)

2

For a $3\times3$ matrix $M$, let $\operatorname{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ …

Topic: JEE Main 2025 (22 January Evening Shift)

3

If $x=f(y)$ is the solution of the differential equation $(1+y^{2})+\big(x-2e^{\tan^{-1}y}\big)\dfrac{dy}{dx}=0,\ y\in\…

Topic: JEE Main 2025 (22 January Evening Shift)

4

The perpendicular distance of the line $\dfrac{x-1}{2}=\dfrac{y+2}{-1}=\dfrac{z+3}{2}$ from the point $P(2,-10,1)$ is:

Topic: JEE Main 2025 (22 January Evening Shift)

5

If $A$ and $B$ are two events such that $P(A \cap B)=0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equ…

Topic: JEE Main 2025 (22 January Evening Shift)

6

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion o…

Topic: JEE Main 2025 (22 January Evening Shift)

7

The sum of all values of $\theta \in [0,2\pi]$ satisfying $2\sin^2\theta=\cos 2\theta$ and $2\cos^2\theta=3\sin\theta$ …

Topic: JEE Main 2025 (22 January Evening Shift)

8

If the system of linear equations : $\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{az}=\mathrm{a}+1 \\ & -x-3 y+\mat…

Topic: JEE Main 2025 (22 January Evening Shift)

9

Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3\hat i+2\hat j+2\hat …

Topic: JEE Main 2025 (22 January Evening Shift)

10

Let \alpha_\theta and \beta_\theta be the distinct roots of $2x^2+(\cos\theta)x-1=0$, $\theta\in(0,2\pi)$. If $m$ and $…

Topic: JEE Main 2025 (22 January Evening Shift)

1

The area of the region enclosed by the curves $y=x^2-4 x+4$ and $y^2=16-8 x$ is :

Topic: JEE Main 2025 (22 January Evening Shift)

2

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys …

Topic: JEE Main 2025 (22 January Evening Shift)

3

If $\displaystyle \lim_{x\to\infty}\left(\frac{e}{1-e}\left(\frac{1}{e}-\frac{x}{1+x}\right)\right)^{x}=\alpha$, then t…

Topic: JEE Main 2025 (22 January Evening Shift)

4

Let $f(x)=\displaystyle \int_{0}^{e^{x^{2}}}\frac{t^{2}-8t+15}{e^{t}}\,dt,\ x\in\mathbb{R}$. Then the numbers of loca…

Topic: JEE Main 2025 (22 January Evening Shift)

5

Let $A=\{1,2,3,4\}$ and $B=\{1,4,9,16\}$. Then the number of many-one functions $f:A\to B$ such that $1\in f(A)$ is equ…

Topic: JEE Main 2025 (22 January Evening Shift)

6

Let the curve $z(1+i)+\overline{z}(1-i)=4,\ z\in\mathbb{C}$, divide the region $|z-3|\le 1$ into two parts of areas $\a…

Topic: JEE Main 2025 (22 January Evening Shift)

7

Suppose that the number of terms in an A.P. is $2k$, $k\in\mathbb{N}$. If the sum of all odd terms of the A.P. is $40$,…

Topic: JEE Main 2025 (22 January Evening Shift)

8

Let $P(4,4\sqrt{3})$ be a point on the parabola $y^{2}=4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ a…

Topic: JEE Main 2025 (22 January Evening Shift)

9

Let $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$ and $H:\ \dfrac{x^{2}}{A^{2}}-\dfrac{y^{2}}{B^{2}}=1$. Let …

Topic: JEE Main 2025 (22 January Evening Shift)