JEE (Advanced) 2024 Paper-2

Considering only the principal values of the inverse trigonometric functions, the value of $$\tan\left(\sin^{-1}\left(\frac{3}{5}\right) - 2\cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$$ is

Let $$S = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x \geq 0, y \geq 0, y^2 \leq 4x, y^2 \leq 12 - 2x \text{ and } 3y + \sqrt{8}x \leq 5\sqrt{8}\}$$. If the area of the region $$S$$ is $$\alpha\sqrt{2}$$, then $$\alpha$$ is equal to

Let $$k \in \mathbb{R}$$. If $$\lim_{x \to 0^+} \left(\sin(\sin kx) + \cos x + x\right)^{\frac{2}{x}} = e^6$$, then the value of $$k$$ is

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by

$$f(x) = \begin{cases} x^2 \sin\left(\frac{\pi}{x^2}\right), & \text{if } x \neq 0, \\ 0, & \text{if } x = 0. \end{cases}$$

Then which of the following statements is TRUE?

Let $$S$$ be the set of all $$(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$$ such that

$$\lim_{x \to \infty} \frac{\sin(x^2)(\log_e x)^\alpha \sin\left(\frac{1}{x^2}\right)}{x^{\alpha\beta}(\log_e(1+x))^\beta} = 0$$

Then which of the following is (are) correct?

A straight line drawn from the point $$P(1,3,2)$$, parallel to the line $$\frac{x-2}{1} = \frac{y-4}{2} = \frac{z-6}{1}$$, intersects the plane $$L_1 : x - y + 3z = 6$$ at the point $$Q$$. Another straight line which passes through $$Q$$ and is perpendicular to the plane $$L_1$$ intersects the plane $$L_2 : 2x - y + z = -4$$ at the point $$R$$. Then which of the following statements is(are) TRUE?

Let $$A_1$$, $$B_1$$, $$C_1$$ be three points in the $$xy$$-plane. Suppose that the lines $$A_1C_1$$ and $$B_1C_1$$ are tangents to the curve $$y^2 = 8x$$ at $$A_1$$ and $$B_1$$, respectively. If $$O = (0,0)$$ and $$C_1 = (-4, 0)$$, then which of the following statements is (are) TRUE?

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function such that $$f(x+y) = f(x) + f(y)$$ for all $$x, y \in \mathbb{R}$$, and $$g : \mathbb{R} \to (0, \infty)$$ be a function such that $$g(x+y) = g(x)g(y)$$ for all $$x, y \in \mathbb{R}$$. If $$f\left(\frac{-3}{5}\right) = 12$$ and $$g\left(\frac{-1}{3}\right) = 2$$, then the value of $$\left(f\left(\frac{1}{4}\right) + g(-2) - 8\right)g(0)$$ is ______.

A bag contains $$N$$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $$i = 1, 2, 3$$, let $$W_i$$, $$G_i$$ and $$B_i$$ denote the events that the ball drawn in the $$i^{th}$$ draw is a white ball, green ball, and blue ball, respectively. If the probability $$P(W_1 \cap G_2 \cap B_3) = \frac{2}{5N}$$ and the conditional probability $$P(B_3 \mid W_1 \cap G_2) = \frac{2}{9}$$, then $$N$$ equals ______.

Let the function $$f : \mathbb{R} \to \mathbb{R}$$ be defined by

$$f(x) = \frac{\sin x}{e^{\pi x}} \cdot \frac{(x^{2023} + 2024x + 2025)}{(x^2 - x + 3)} + \frac{2}{e^{\pi x}} \cdot \frac{(x^{2023} + 2024x + 2025)}{(x^2 - x + 3)}$$.

Then the number of solutions of $$f(x) = 0$$ in $$\mathbb{R}$$ is ______.