JEE (Advanced) 2023 Paper-2

Let $$f: [1, \infty) \to \mathbb{R}$$ be a differentiable function such that $$f(1) = \frac{1}{3}$$ and $$3\int_1^x f(t)\,dt = xf(x) - \frac{x^3}{3}$$, $$x \in [1, \infty)$$. Let $$e$$ denote the base of the natural logarithm. Then the value of $$f(e)$$ is

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $$\frac{1}{3}$$, then the probability that the experiment stops with head is

For any $$y \in \mathbb{R}$$, let $$\cot^{-1}(y) \in (0, \pi)$$ and $$\tan^{-1}(y) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$. Then the sum of all the solutions of the equation $$\tan^{-1}\left(\frac{6y}{9 - y^2}\right) + \cot^{-1}\left(\frac{9 - y^2}{6y}\right) = \frac{2\pi}{3}$$ for $$0 < |y| < 3$$, is equal to

Let the position vectors of the points P, Q, R and S be $$\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$$, $$\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$$, $$\vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}$$ and $$\vec{d} = 2\hat{i} + \hat{j} + \hat{k}$$, respectively. Then which of the following statements is true?

Let $$M = (a_{ij})$$, $$i, j \in \{1, 2, 3\}$$, be the $$3 \times 3$$ matrix such that $$a_{ij} = 1$$ if $$j + 1$$ is divisible by $$i$$, otherwise $$a_{ij} = 0$$. Then which of the following statements is (are) true?

Let $$f : (0,1) \to \mathbb{R}$$ be the function defined as $$f(x) = [4x]\left(x - \frac{1}{4}\right)^2\left(x - \frac{1}{2}\right)$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. Then which of the following statements is(are) true?

Let S be the set of all twice differentiable functions $$f$$ from $$\mathbb{R}$$ to $$\mathbb{R}$$ such that $$\frac{d^2f}{dx^2}(x) > 0$$ for all $$x \in (-1, 1)$$. For $$f \in S$$, let $$X_f$$ be the number of points $$x \in (-1, 1)$$ for which $$f(x) = x$$. Then which of the following statements is(are) true?

For $$x \in \mathbb{R}$$, let $$\tan^{-1}(x) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$. Then the minimum value of the function $$f : \mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = \int_0^{x\tan^{-1}x} \frac{e^{(t - \cos t)}}{1 + t^{2023}}\,dt$$ is

For $$x \in \mathbb{R}$$, let $$y(x)$$ be a solution of the differential equation $$(x^2 - 5)\frac{dy}{dx} - 2xy = -2x(x^2 - 5)^2$$ such that $$y(2) = 7$$. Then the maximum value of the function $$y(x)$$ is

Let X be the set of all five digit numbers formed using 1, 2, 2, 2, 4, 4, 0. For example, 22240 is in X while 02244 and 44422 are not in X. Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of 38p is equal to