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 ______.

Let $$\vec{p} = 2\hat{i} + \hat{j} + 3\hat{k}$$ and $$\vec{q} = \hat{i} - \hat{j} + \hat{k}$$. If for some real numbers $$\alpha$$, $$\beta$$ and $$\gamma$$, we have $$15\hat{i} + 10\hat{j} + 6\hat{k} = \alpha(2\vec{p} + \vec{q}) + \beta(\vec{p} - 2\vec{q}) + \gamma(\vec{p} \times \vec{q})$$, then the value of $$\gamma$$ is ______.

A normal with slope $$\frac{1}{\sqrt{6}}$$ is drawn from the point $$(0, -\alpha)$$ to the parabola $$x^2 = -4ay$$, where $$a > 0$$. Let $$L$$ be the line passing through $$(0, -\alpha)$$ and parallel to the directrix of the parabola. Suppose that $$L$$ intersects the parabola at two points $$A$$ and $$B$$. Let $$r$$ denote the length of the latus rectum and $$s$$ denote the square of the length of the line segment $$AB$$. If $$r : s = 1 : 16$$, then the value of $$24a$$ is ______.

Let the function $$f : [1, \infty) \to \mathbb{R}$$ be defined by

$$f(t) = \begin{cases} (-1)^{n+1} \cdot 2, & \text{if } t = 2n-1, \, n \in \mathbb{N}, \\ \frac{(2n+1-t)}{2} f(2n-1) + \frac{(t-(2n-1))}{2} f(2n+1), & \text{if } 2n-1 < t < 2n+1, \, n \in \mathbb{N}. \end{cases}$$

Define $$g(x) = \int_1^x f(t) \, dt$$, $$x \in (1, \infty)$$. Let $$\alpha$$ denote the number of solutions of the equation $$g(x) = 0$$ in the interval $$(1, 8]$$ and $$\beta = \lim_{x \to 1^+} \frac{g(x)}{x-1}$$. Then the value of $$\alpha + \beta$$ is equal to ______.

If $$n(X) = {}^{m}C_6$$, then the value of $$m$$ is ______.

If the value of $$n(Y) + n(Z)$$ is $$k^2$$, then $$|k|$$ is ______.

The value of $$2\int_0^{\frac{\pi}{2}} f(x)g(x) \, dx - \int_0^{\frac{\pi}{2}} g(x) \, dx$$ is ______.

The value of $$\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x)g(x) \, dx$$ is ______.

A region in the form of an equilateral triangle (in $$x - y$$ plane) of height $$L$$ has a uniform magnetic field $$\vec{B}$$ pointing in the $$+z$$-direction. A conducting loop $$PQR$$, in the form of an equilateral triangle of the same height $$L$$, is placed in the $$x - y$$ plane with its vertex $$P$$ at $$x = 0$$ in the orientation shown in the figure. At $$t = 0$$, the loop starts entering the region of the magnetic field with a uniform velocity $$\vec{v}$$ along the $$+x$$-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Which of the following graph best depicts the variation of the induced emf ($$E$$) in the loop as a function of the distance ($$x$$) starting from $$x = 0$$?

A particle of mass $$m$$ is under the influence of the gravitational field of a body of mass $$M$$ ($$\gg m$$). The particle is moving in a circular orbit of radius $$r_0$$ with time period $$T_0$$ around the mass $$M$$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $$V_c(r) = m\alpha / r^3$$, where $$\alpha$$ is a positive constant of suitable dimensions and $$r$$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $$r_0$$ in the combined gravitational potential due to $$M$$ and $$V_c(r)$$, but with a new time period $$T_1$$, then $$(T_1^2 - T_0^2)/T_1^2$$ is given by

[$$G$$ is the gravitational constant.]

A metal target with atomic number $$Z = 46$$ is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio $$r$$ of the wavelengths of the $$K_\alpha$$-line and the cut-off is found to be $$r = 2$$. If the same electron beam bombards another metal target with $$Z = 41$$, the value of $$r$$ will be

A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $$m$$ and radius $$r$$ and it is in a uniform vertical magnetic field $$B_0$$, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $$g$$, on two conducting supports at $$P$$ and $$Q$$. When a current $$I$$ is passed through the loop, the loop turns about the line $$PQ$$ by an angle $$\theta$$ given by

A small electric dipole $$\vec{p}_0$$, having a moment of inertia $$I$$ about its center, is kept at a distance $$r$$ from the center of a spherical shell of radius $$R$$. The surface charge density $$\sigma$$ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $$\theta$$ as shown in the figure. While staying at a distance $$r$$, the dipole is free to rotate about its center.

If released from rest, then which of the following statement(s) is (are) correct?

[$$\varepsilon_0$$ is the permittivity of free space.]

A table tennis ball has radius $$(3/2) \times 10^{-2}$$ m and mass $$(22/7) \times 10^{-3}$$ kg. It is slowly pushed down into a swimming pool to a depth of $$d = 0.7$$ m below the water surface and then released from rest. It emerges from the water surface at speed $$v$$, without getting wet, and rises up to a height $$H$$. Which of the following option(s) is (are) correct?

[Given: $$\pi = 22/7$$, $$g = 10$$ ms$$^{-2}$$, density of water $$= 1 \times 10^3$$ kg m$$^{-3}$$, viscosity of water $$= 1 \times 10^{-3}$$ Pa·s.]

A positive, singly ionized atom of mass number $$A_M$$ is accelerated from rest by the voltage 192 V. Thereafter, it enters a rectangular region of width $$w$$ with magnetic field $$\vec{B}_0 = 0.1\hat{k}$$ Tesla, as shown in the figure. The ion finally hits a detector at the distance $$x$$ below its starting trajectory.

[Given: Mass of neutron/proton $$= (5/3) \times 10^{-27}$$ kg, charge of the electron $$= 1.6 \times 10^{-19}$$ C.]

Which of the following option(s) is(are) correct?

The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm. The maximum percentage error in the determination of the volume is ______.

A ball is thrown from the location $$(x_0, y_0) = (0,0)$$ of a horizontal playground with an initial speed $$v_0$$ at an angle $$\theta_0$$ from the $$+x$$-direction. The ball is to be hit by a stone, which is thrown at the same time from the location $$(x_1, y_1) = (L, 0)$$. The stone is thrown at an angle $$(180^\circ - \theta_1)$$ from the $$+x$$-direction with a suitable initial speed. For a fixed $$v_0$$, when $$(\theta_0, \theta_1) = (45^\circ, 45^\circ)$$, the stone hits the ball after time $$T_1$$, and when $$(\theta_0, \theta_1) = (60^\circ, 30^\circ)$$, it hits the ball after time $$T_2$$. In such a case, $$(T_1/T_2)^2$$ is ________.

A charge is kept at the central point $$P$$ of a cylindrical region. The two edges subtend a half-angle $$\theta$$ at $$P$$, as shown in the figure. When $$\theta = 30^\circ$$, then the electric flux through the curved surface of the cylinder is $$\Phi$$. If $$\theta = 60^\circ$$, then the electric flux through the curved surface becomes $$\Phi / \sqrt{n}$$, where the value of $$n$$ is ________.

Two equilateral-triangular prisms $$P_1$$ and $$P_2$$ are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism $$P_1$$ at an angle of incidence $$\theta$$ such that the outgoing ray undergoes minimum deviation in prism $$P_2$$. If the respective refractive indices of $$P_1$$ and $$P_2$$ are $$\sqrt{\dfrac{3}{2}}$$ and $$\sqrt{3}$$, $$\theta = \sin^{-1}\left[\sqrt{\dfrac{3}{2}} \sin\left(\dfrac{\pi}{\beta}\right)\right]$$, where the value of $$\beta$$ is ________.

An infinitely long thin wire, having a uniform charge density per unit length of 5 nC/m, is passing through a spherical shell of radius 1 m, as shown in the figure. A 10 nC charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points $$P$$ and $$R$$, in Volt, is ________.

[Given: In SI units $$\dfrac{1}{4\pi \varepsilon_0} = 9 \times 10^9$$, ln 2 = 0.7. Ignore the area pierced by the wire.]

A spherical soap bubble inside an air chamber at pressure $$P_0 = 10^5$$ Pa has a certain radius so that the excess pressure inside the bubble is $$\Delta P = 144$$ Pa. Now, the chamber pressure is reduced to $$8P_0/27$$ so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure $$\Delta P$$ in both the cases to be much smaller than the chamber pressure. The new excess pressure $$\Delta P$$ in Pa is ________.

The 8th bright fringe above the point $$O$$ oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer ($$\mu$$m), is ________.

The maximum speed in $$\mu$$ m/s at which the $$8^{th}$$ bright fringe will move is ________.

If the collision occurs at time $$t_0 = 0$$, the value of $$v_{cm}/(a\omega)$$ will be ________.

If the collision occurs at time $$t_0 = \pi/(2\omega)$$, then the value of $$4b^2/a^2$$ will be ________.

According to Bohr's model, the highest kinetic energy is associated with the electron in the

In a metal deficient oxide sample, $$M_XY_2O_4$$ (M and Y are metals), M is present in both +2 and +3 oxidation states and Y is in +3 oxidation state. If the fraction of $$M^{2+}$$ ions present in M is $$\frac{1}{3}$$, the value of X is ______.

In the following reaction sequence, the major product Q is

The species formed on fluorination of phosphorus pentachloride in a polar organic solvent are

An aqueous solution of hydrazine ($$N_2H_4$$) is electrochemically oxidized by $$O_2$$, thereby releasing chemical energy in the form of electrical energy. One of the products generated from the electrochemical reaction is $$N_2(g)$$.

Choose the correct statement(s) about the above process

The option(s) with correct sequence of reagents for the conversion of P to Q is(are)

The compound(s) having peroxide linkage is(are)

To form a complete monolayer of acetic acid on 1 g of charcoal, 100 mL of 0.5 M acetic acid was used. Some of the acetic acid remained unadsorbed. To neutralize the unadsorbed acetic acid, 40 mL of 1 M NaOH solution was required. If each molecule of acetic acid occupies $$P \times 10^{-23}$$ $$m^2$$ surface area on charcoal, the value of P is ________.

[Use given data : Surface area of charcoal = $$1.5 \times 10^2$$ $$m^2 g^{-1}$$; Avogadro's number ($$N_A$$) = $$6.0 \times 10^{23}$$ $$mol^{-1}$$]

Vessel-1 contains $$w_2$$ g of a non-volatile solute X dissolved in $$w_1$$ g of water. Vessel-2 contains $$w_2$$ g of another non-volatile solute Y dissolved in $$w_1$$ g of water. Both the vessels are at the same temperature and pressure. The molar mass of X is 80% of that of Y. The van't Hoff factor for X is 1.2 times of that of Y for their respective concentrations.

The elevation of boiling point for solution in Vessel-1 is ______ % of the solution in Vessel-2.

For a double strand DNA, one strand is given below:

The amount of energy required to split the double strand DNA into two single strands is ______ kcal $$mol^{-1}$$.

[Given: Average energy per H-bond for A-T base pair = 1.0 kcal $$mol^{-1}$$, G-C base pair = 1.5 kcal $$mol^{-1}$$, and A-U base pair = 1.25 kcal $$mol^{-1}$$. Ignore electrostatic repulsion between the phosphate groups.]

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated.

When the age of the sample is $$P \times 10^8$$ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is ______.

[Given : Half-life of U-238 is $$4.5 \times 10^9$$ years; $$\log_e 2$$ = 0.693]

Among $$[Co(CN)_4]^{4-}$$, $$[Co(CO)_3(NO)]$$, $$XeF_4$$, $$[PCl_4]^+$$, $$[PdCl_4]^{2-}$$, $$[ICl_4]^-$$, $$[Cu(CN)_4]^{3-}$$ and $$P_4$$ the total number of species with tetrahedral geometry is ______.

An organic compound P having molecular formula $$C_6H_6O_3$$ gives ferric chloride test and does not have intramolecular hydrogen bond. The compound P reacts with 3 equivalents of $$NH_2OH$$ to produce oxime Q. Treatment of P with excess methyl iodide in the presence of KOH produces compound R as the major product. Reaction of R with excess iso-butylmagnesium bromide followed by treatment with $$H_3O^+$$ gives compound S as the major product.

The total number of methyl ($$-CH_3$$) group(s) in compound S is ______.

Sum of number of oxygen atoms in S and T is ______.

The molecular weight of U is ______.

The number of moles of potassium iodide required to produce two moles of P is ______.

The number of zinc ions present in the molecular formula of Q is ______.