NTA JEE Main 2025 April 4th Shift 1

Let $$f, g : (1, \infty) \to \mathbb{R}$$ be defined as $$f(x) = \dfrac{2x + 3}{5x + 2}$$ and $$g(x) = \dfrac{2 - 3x}{1 - x}$$. If the range of the function $$f \circ g : [2, 4] \to \mathbb{R}$$ is $$[\alpha, \beta]$$, then $$\dfrac{1}{\beta - \alpha}$$ is equal to

Consider the sets $$A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$$, $$B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$$, $$C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \le 4\}$$, and $$D = A \cap B$$. The total number of one-one functions from the set D to the set C is:

Let $$A = \{1, 6, 11, 16, \ldots\}$$ and $$B = \{9, 16, 23, 30, \ldots\}$$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $$n(A \cup B)$$ is

For an integer $$n \ge 2$$, if the arithmetic mean of all coefficients in the binomial expansion of $$(x + y)^{2n-3}$$ is 16, then the distance of the point $$P(2n - 1, n^2 - 4n)$$ from the line $$x + y = 8$$ is:

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:

Let the shortest distance between the lines $$\dfrac{x - 3}{3} = \dfrac{y - \alpha}{-1} = \dfrac{z - 3}{1}$$ and $$\dfrac{x + 3}{-3} = \dfrac{y + 7}{2} = \dfrac{z - \beta}{4}$$ be $$3\sqrt{30}$$. Then the positive value of $$5\alpha + \beta$$ is

If $$\displaystyle\lim_{x \to 1} \dfrac{(x - 1)(6 + \lambda \cos(x - 1)) + \mu \sin(1 - x)}{(x - 1)^3} = -1$$, where $$\lambda, \mu \in \mathbb{R}$$, then $$\lambda + \mu$$ is equal to

Let $$f : [0, \infty) \to \mathbb{R}$$ be differentiable function such that $$f(x) = 1 - 2x + \displaystyle\int_0^x e^{x-t} f(t) \, dt$$ for all $$x \in [0, \infty)$$. Then the area of the region bounded by $$y = f(x)$$ and the coordinate axes is

Let A and B be two distinct points on the line $$L : \dfrac{x - 6}{3} = \dfrac{y - 7}{2} = \dfrac{z - 7}{-2}$$. Both A and B are at a distance $$2\sqrt{17}$$ from the foot of perpendicular drawn from the point $$(1, 2, 3)$$ on the line L. If O is the origin, then $$\vec{OA} \cdot \vec{OB}$$ is equal to:

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a continuous function satisfying $$f(0) = 1$$ and $$f(2x) - f(x) = x$$ for all $$x \in \mathbb{R}$$. If $$\displaystyle\lim_{n \to \infty} \left\{f(x) - f\left(\dfrac{x}{2^n}\right)\right\} = G(x)$$, then $$\displaystyle\sum_{r=1}^{10} G(r^2)$$ is equal to