NTA JEE Main 2025 April 8th Shift 2

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

A line passing through the point P(a, $$\theta$$) makes an acute angle $$\alpha$$ with the positive x-axis. Let this line be rotated about the point P through an angle $$\frac{\alpha}{2}$$ in the clock-wise direction. If in the new position, the slope of the line is $$2 - \sqrt{3}$$ and its distance from the origin is $$\frac{1}{\sqrt{2}}$$, then the value of $$3a^2\tan^2\alpha - 2\sqrt{3}$$ is

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

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

Let $$A = \{0, 1, 2, 3, 4, 5\}$$. Let R be a relation on A defined by $$(x, y) \in R$$ if and only if $$\max\{x, y\} \in \{3, 4\}$$. Then among the statements
$$(S_1)$$ : The number of elements in R is 18, and
$$(S_2)$$ : The relation R is symmetric but neither reflexive nor transitive

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

Let $$f(x) = x - 1$$ and $$g(x) = e^x$$ for $$x \in \mathbb{R}$$. If $$\frac{dy}{dx} = \left(e^{-2\sqrt{x}} g(f(f(x))) - \frac{y}{\sqrt{x}}\right)$$, $$y(0) = 0$$, then $$y(1)$$ is :-

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

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 $$\det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n$$, $$m, n \in \mathbb{N}$$, then $$m + n$$ is equal to

Given below are two statements :
Statement I : $$\lim_{x \to 0} \left(\frac{\tan^{-1}x + \log_{e}\sqrt{\frac{1+x}{1-x}} - 2x}{x^5}\right) = \frac{2}{5}$$
Statement II : $$\lim_{x \to 1} \left(x^{\frac{2}{1-x}}\right) = \frac{1}{e^2}$$
In the light of the above statements, choose the correct answer :