JEE (Advanced) 2024 Paper-1

Let $$f(x)$$ be a continuously differentiable function on the interval $$(0, \infty)$$ such that $$f(1) = 2$$ and $$\lim_{t \to x} \frac{t^{10}f(x) - x^{10}f(t)}{t^9 - x^9} = 1$$ for each $$x \gt 0$$. Then, for all $$x \gt 0$$, $$f(x)$$ is equal to

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is $$\frac{1}{2}$$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $$\frac{1}{6}$$. Then the probability that the student knows the answer of a randomly chosen question is

Let $$\frac{\pi}{2} \lt x \lt \pi$$ be such that $$\cot x = \frac{-5}{\sqrt{11}}$$. Then $$\left(\sin \frac{11x}{2}\right)(\sin 6x - \cos 6x) + \left(\cos \frac{11x}{2}\right)(\sin 6x + \cos 6x)$$ is equal to

Consider the ellipse $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Let $$S(p, q)$$ be a point in the first quadrant such that $$\frac{p^2}{9} + \frac{q^2}{4} \gt 1$$. Two tangents are drawn from $$S$$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $$T$$ in the fourth quadrant. Let $$R$$ be the vertex of the ellipse with positive $$x$$-coordinate and $$O$$ be the center of the ellipse. If the area of the triangle $$\triangle ORT$$ is $$\frac{3}{2}$$, then which of the following options is correct?

Let $$S = \{a + b\sqrt{2} : a, b \in \mathbb{Z}\}$$, $$T_1 = \{(-1 + \sqrt{2})^n : n \in \mathbb{N}\}$$ and $$T_2 = \{(1 + \sqrt{2})^n : n \in \mathbb{N}\}$$. Then which of the following statements is (are) TRUE?

Let $$\mathbb{R}^2$$ denote $$\mathbb{R} \times \mathbb{R}$$. Let $$S = \{(a, b, c) : a, b, c \in \mathbb{R}$$ and $$ax^2 + 2bxy + cy^2 \gt 0$$ for all $$(x, y) \in \mathbb{R}^2 - \{(0, 0)\}\}$$. Then which of the following statements is (are) TRUE?

Let $$\mathbb{R}^3$$ denote the three-dimensional space. Take two points $$P = (1, 2, 3)$$ and $$Q = (4, 2, 7)$$. Let $$dist(X, Y)$$ denote the distance between two points $$X$$ and $$Y$$ in $$\mathbb{R}^3$$. Let $$S = \{X \in \mathbb{R}^3 : (dist(X, P))^2 - (dist(X, Q))^2 = 50\}$$ and $$T = \{Y \in \mathbb{R}^3 : (dist(Y, Q))^2 - (dist(Y, P))^2 = 50\}$$. Then which of the following statements is (are) TRUE?

Let $$a = 3\sqrt{2}$$ and $$b = \frac{1}{5^{1/6}\sqrt{6}}$$. If $$x, y \in \mathbb{R}$$ are such that $$3x + 2y = \log_a (18)^{5/4}$$ and $$2x - y = \log_b (\sqrt{1080})$$, then $$4x + 5y$$ is equal to ________.

Let $$f(x) = x^4 + ax^3 + bx^2 + c$$ be a polynomial with real coefficients such that $$f(1) = -9$$. Suppose that $$i\sqrt{3}$$ is a root of the equation $$4x^3 + 3ax^2 + 2bx = 0$$, where $$i = \sqrt{-1}$$. If $$\alpha_1, \alpha_2, \alpha_3$$, and $$\alpha_4$$ are all the roots of the equation $$f(x) = 0$$, then $$|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$$ is equal to ________.

Let $$S = \left\{A = \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\}\right\}$$, where $$|A|$$ denotes the determinant of $$A$$. Then the number of elements in $$S$$ is ________.