JEE (Advanced) 2023 Paper-1

Let $$S = (0, 1) \cup (1, 2) \cup (3, 4)$$ and $$T = \{0, 1, 2, 3\}$$. Then which of the following statements is(are) true?

Let $$T_1$$ and $$T_2$$ be two distinct common tangents to the ellipse $$E: \frac{x^2}{6} + \frac{y^2}{3} = 1$$ and the parabola $$P: y^2 = 12x$$. Suppose that the tangent $$T_1$$ touches P and E at the points $$A_1$$ and $$A_2$$, respectively and the tangent $$T_2$$ touches P and E at the points $$A_4$$ and $$A_3$$, respectively. Then which of the following statements is(are) true?

Let $$f : [0, 1] \to [0, 1]$$ be the function defined by $$f(x) = \frac{x^3}{3} - x^2 + \frac{5}{9}x + \frac{17}{36}$$. Consider the square region $$S = [0, 1] \times [0, 1]$$. Let $$G = \{(x, y) \in S : y > f(x)\}$$ be called the green region and $$R = \{(x, y) \in S : y < f(x)\}$$ be called the red region. Let $$L_h = \{(x, h) \in S : x \in [0, 1]\}$$ be the horizontal line drawn at a height $$h \in [0, 1]$$. Then which of the following statements is(are) true?

Let $$f : (0, 1) \to \mathbb{R}$$ be the functions defined as $$f(x) = \sqrt{n}$$ if $$x \in \left[\frac{1}{n+1}, \frac{1}{n}\right)$$ where $$n \in \mathbb{N}$$. Let $$g : (0, 1) \to \mathbb{R}$$ be a function such that $$\int_{x^2}^{x} \sqrt{\frac{1-t}{t}} \, dt < g(x) < 2\sqrt{x}$$ for all $$x \in (0, 1)$$. Then $$\lim_{x \to 0} f(x)g(x)$$

Let Q be the cube with the set of vertices $$\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \in \{0, 1\}\}$$. Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0, 0, 0) and (1, 1, 1) is in S. For lines $$\ell_1$$ and $$\ell_2$$, let $$d(\ell_1, \ell_2)$$ denote the shortest distance between them. Then the maximum value of $$d(\ell_1, \ell_2)$$, as $$\ell_1$$ varies over F and $$\ell_2$$ varies over S, is

Let $$X = \left\{(x, y) \in \mathbb{Z} \times \mathbb{Z} : \frac{x^2}{8} + \frac{y^2}{20} < 1 \text{ and } y^2 < 5x\right\}$$. Three distinct points P, Q and R are randomly chosen from X. Then the probability that P, Q and R form a triangle whose area is a positive integer, is

Let P be a point on the parabola $$y^2 = 4ax$$, where $$a > 0$$. The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ, where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (a, m) is

Let $$\tan^{-1}(x) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, for $$x \in \mathbb{R}$$. Then the number of real solutions of the equation $$\sqrt{1 + \cos(2x)} = \sqrt{2} \tan^{-1}(\tan x)$$ in the set $$\left(-\frac{3\pi}{2}, \frac{\pi}{2}\right) \cup \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$$ is equal to

Let $$n \geq 2$$ be a natural number and $$f : [0, 1] \to \mathbb{R}$$ be the function defined by

$$f(x) = \begin{cases} n(1 - 2nx) & \text{if } 0 \leq x \leq \frac{1}{2n} \\ 2n(2nx - 1) & \text{if } \frac{1}{2n} \leq x \leq \frac{3}{4n} \\ 4n(1 - nx) & \text{if } \frac{3}{4n} \leq x \leq \frac{1}{n} \\ \frac{n}{n-1}(nx - 1) & \text{if } \frac{1}{n} \leq x \leq 1 \end{cases}$$

If n is such that the area of the region bounded by the curves $$x = 0$$, $$x = 1$$, $$y = 0$$ and $$y = f(x)$$ is 4, then the maximum value of the function f is

Let $$\overset{r}{75...57}$$ denote the $$(r + 2)$$ digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum $$S = 77 + 757 + 7557 + \cdots + \overset{98}{75...57}$$. If $$S = \frac{\overset{99}{75...57} + m}{n}$$, where m and n are natural numbers less than 3000, then the value of $$m + n$$ is