Let $A = {\theta \in [0,2\pi] : 1 + 10,\mathrm{Re}\left(\dfrac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta}\right) = 0}$. Then $\displaystyle \sum_{\theta \in A} \theta^2$ is equal to

1

The value of $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \ta…

Topic: JEE Main 2025 (8 April Evening Shift)

2

The sum of the squares of the roots of $|x - 2|^2 + |x - 2| - 2 = 0$ and the squares of the roots of $x^2 - 2|x - 3| - …

Topic: JEE Main 2025 (8 April Evening Shift)

3

Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r…

Topic: JEE Main 2025 (8 April Evening Shift)

4

A line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive x-axis. Let this line be rot…

Topic: JEE Main 2025 (8 April Evening Shift)

5

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $\alpha $ with…

Topic: JEE Main 2025 (8 April Evening Shift)

6

If $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty= \frac{\pi^4}{90} $, $\frac{1}{1^4} + \frac{1}{3^4} …

Topic: JEE Main 2025 (8 April Evening Shift)

7

Let $f(x)=x-1$ and $g(x)=e^x$ for $x\in\mathbb{R}$. If $\dfrac{dy}{dx}=e^{-2\sqrt{x}}g\big(f(f(x))\big)-\dfrac{y}{\sqrt…

Topic: JEE Main 2025 (8 April Evening Shift)

8

Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-…

Topic: JEE Main 2025 (8 April Evening Shift)

1

Given below are two statements: Statement I: $\displaystyle \lim_{x \to 0} \left( \tan^{-1}x + \log_e \dfrac{\sqrt{1+x…

Topic: JEE Main 2025 (8 April Evening Shift)

2

The integral $\displaystyle \int_{-1}^{\tfrac{3}{2}} \left( |\pi^2 x \sin(\pi x)| \right) dx$ is equal to

Topic: JEE Main 2025 (8 April Evening Shift)

3

Let $ A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix} $.If $ \det(\text…

Topic: JEE Main 2025 (8 April Evening Shift)

4

Let $A={0,1,2,3,4,5}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ iff $\max{x,y}\in{3,4}$. Then among the sta…

Topic: JEE Main 2025 (8 April Evening Shift)

5

Let $f(x)$ be a positive function and $I_{1}=\int_{-\tfrac{1}{2}}^{1} 2x,f\left(2x(1-2x)\right),dx$ and $I_{2}=\int_{-1…

Topic: JEE Main 2025 (8 April Evening Shift)

6

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. T…

Topic: JEE Main 2025 (8 April Evening Shift)

7

If $A$ and $B$ are two events such that $P(A)=0.7,\ P(B)=0.4$ and $P(A\cap \overline{B})=0.5$, where $\overline{B}$ den…

Topic: JEE Main 2025 (8 April Evening Shift)

8

Let $\vec a=\hat i+2\hat j+\hat k$ and $\vec b=2\hat i+\hat j-\hat k$. Let $\vec c$ be a unit vector in the plane of th…

Topic: JEE Main 2025 (8 April Evening Shift)

9

Let the function $f(x)=\dfrac{x}{3}+\dfrac{3}{x}+3,\ x\ne0$ be strictly increasing in $(-\infty,\alpha_1)\cup(\alpha_2,…

Topic: JEE Main 2025 (8 April Evening Shift)

10

The number of integral terms in the expansion of $\left(5^{\tfrac{1}{2}}+7^{\tfrac{1}{8}}\right)^{1016}$ is:

Topic: JEE Main 2025 (8 April Evening Shift)