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

Let $k \in \mathbb{R}$. If $\lim _{x \rightarrow 0^{+}}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^{6}$, then the value of $k$ is

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

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 $\lim _{x \rightarrow \infty} \frac{\sin \left(x^{2}\right)\left(\log _{e} x\right)^{\alpha} \sin \left(\frac{1}{x^{2}}\right)}{x^{\alpha \beta}\left(\log _{e}(1+x)\right)^{\beta}}=0 .$ 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 $L_{1}: x-y+3 z=6$ at the point $Q$. Another straight line which passesthrough $Q$ and is perpendicular to the plane $L_{1}$ intersects the plane $L_{2}: 2 x-y+z=-4$ 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 $x y$-plane. Suppose that the lines $A_{1} C_{1}$ and $B_{1} C_{1}$ are tangents to the curve $y^{2}=8 x$ 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 $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{R}$, and $g: \mathbb{R} \rightarrow(0, \infty)$ 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 $\qquad$

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 $\mathrm{i}^{\text {th }}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P\left(W_{1} \cap G_{2} \cap B_{3}\right)=\frac{2}{5 N}$ and the conditional probability $P\left(B_{3} \mid W_{1} \cap G_{2}\right)=\frac{2}{9}$, then $N$ equals $\qquad$

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2024}+2024 x+2025\right)}{\left(x^{2}-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2024}+2024 x+2025\right)}{\left(x^{2}-x+3\right)} .$

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

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

then the value of $\gamma$ is $\_\_\_\_\_$

A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha)$ to the parabola $x^{2}=-4$ ay, where $\mathrm{a}>0$. 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 $A B$. If $r: s=1: 16$, then the value of $24 \alpha$ is

A continuous function $f:\left[0, \frac{3}{4}\right] \rightarrow R$ satisfies the equation $f(x)=\left(1+x^{2}\right)\left(1+\int_{0}^{x} \frac{f^{2}(t)}{1+t^{2}} d t\right)$, then $\left[\mathrm{f}\left(\frac{1}{2}\right)\right]$ is: [Note : [K] denotes greatest integer less than or equal to K.]

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 ro with time period $\mathrm{T}_{0}$ around the mass M . Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_{c}(r)=m a / r^{3}$, 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 ro in the combined gravitational potential due to $M$ and $V_{c}(r)$, but with a new time period $T_{1}$, then $\left(T_{1}^{2}-T_{0}^{2} / T_{1}^{2}\right.$ is given by [0pt] [ 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 table tennis ball has radius $(3 / 2) \times 10^{-2} \mathrm{~m}$ and mass $(22 / 7) \times 10^{-3} \mathrm{~kg}$. It is slowly pushed down into a swimming pool to a depth of $\mathrm{d}=0.7 \mathrm{~m}$ 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: $\mathrm{m}=22 / 7, \mathrm{~g}=10 \mathrm{~ms}^{-2}$, density of water $=1 \times 10^{3} \mathrm{kgm}^{-3}$, viscosity of water $=1 \times 10^{-3} \mathrm{~Pa}$-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 $\overrightarrow{\mathrm{B}}_{0}=0.1 \hat{\mathrm{k}}$ 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 $=(5 / 3) \times 10^{-27} \mathrm{~kg}$, charge of the electron $=1.6 \times 10^{-19} \mathrm{C}$.] 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 $\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)=(0,0)$ 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 $\left(x_{1}, y_{1}\right)=(L, 0)$. The stone is thrown at an angle ( $180-\theta_{1}$ ) from the $+x$-direction with a suitable initial speed. For a fixed $v_{0}$, when $\left(\theta_{0}, \theta_{1}\right)=\left(45^{\circ}, 45^{\circ}\right)$, the stone hits the ball after time $\mathrm{T}_{1}$, and when $\left(\theta_{0}, \theta_{1}\right)=\left(60^{\circ}, 30^{\circ}\right)$, it hits the ball after time $\mathrm{T}_{2}$. In such a case, $\left(T_{1} / T_{2}\right)^{2}$ 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 $\frac{\phi}{\sqrt{n}}$, 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 $\sqrt{\frac{3}{2}}$ and $\sqrt{3}$, then $\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]$, where the value of $\beta$ is

An infinitely long thin wire, having a uniform charge density per unit length of $5 \mathrm{nC} / \mathrm{m}$, 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 $\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9}, \ln 2=0.7$. Ignore the area pierced by the wire.]

A spherical soap bubble inside an air chamber at pressure $P_{0}=10^{5} \mathrm{~Pa}$ has a certain radius so that the excess pressure inside the bubble is $\Delta \mathrm{P}=144 \mathrm{~Pa}$. Now, the chamber pressure is reduced to $8 \mathrm{P}_{0} / 27$ 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

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

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

In the following reaction sequence, the major product $\mathbf{Q}$ is

The species formed on fluorination of phosphorus pentachloride in a polar organic solvent 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 $\mathbf{P} \times 10^{-23} \mathrm{~m}^{2}$ surface area on charcoal, the value of $\mathbf{P}$ is $\_\_\_\_\_$ _. [Use given data : Surface area of charcoal $=1.5 \times 10^{2} \mathrm{~m}^{2} \mathrm{~g}^{-1}$; Avogadro's number $\left(N_{A}\right)=6.0 \times 10^{23} \mathrm{~mol}^{-1}$ ]

Vessel-1 contains $\mathbf{w}_{\mathbf{2}} \mathrm{g}$ of a non-volatile solute $\mathbf{X}$ dissolved in $\mathbf{w}_{\mathbf{1}} \mathrm{g}$ of water. Vessel-2 contains $\mathbf{w}_{\mathbf{2}}$ $g$ of another non-volatile solute $\mathbf{Y}$ dissolved in $\mathbf{w}_{1} \mathrm{~g}$ of water. Both the vessels are at the same temperature and pressure. The molar mass of $\mathbf{X}$ is $80 \%$ of that of $\mathbf{Y}$. The van't Hoff factor for $\mathbf{X}$ is 1.2 times of that of $\mathbf{Y}$ 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 $\_\_\_\_\_$ $\mathrm{kcal} \mathrm{mol}^{-1}$. [Given: Average energy per H-bond for A-T base pair $=1.0 \mathrm{kcal} \mathrm{mol}^{-1}$, G-C base pair $=1.5 \mathrm{kcal}$ $\mathrm{mol}^{-1}$, and A-U base pair $=1.25 \mathrm{kcal} \mathrm{mol}^{-1}$. Ignore electrostatic repulsion between the phosphate groups.]

Among $\left[\mathrm{Co}(\mathrm{CN})_{4}\right]^{4-},\left[\mathrm{Co}(\mathrm{CO})_{3}(\mathrm{NO})\right], \mathrm{XeF}_{4},\left[\mathrm{PCl}_{4}\right]^{+},\left[\mathrm{PdCl}_{4}\right]^{2-},[\mathrm{ICl}]^{-},\left[\mathrm{Cu}(\mathrm{CN})_{4}\right]^{3-}$ and $\mathrm{P}_{4}$ the total number of species with tetrahedral geometry is $\_\_\_\_\_$ -.

An organic compound $P$ having molecular formula $\mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}_{3}$ gives ferric chloride test and does not have intramolecular hydrogen bond. The compound $\mathbf{P}$ reacts with 3 equivalents of $\mathrm{NH}_{2} \mathrm{OH}$ to produce oxime $\mathbf{Q}$. Treatment of $\mathbf{P}$ with excess methyl iodide in the presence of KOH produces compound $\mathbf{R}$ as the major product. Reaction of $\mathbf{R}$ with excess iso-butylmagnesium bromide followed by treatment with $\mathrm{H}_{3} \mathrm{O}^{+}$gives compound $\mathbf{S}$ as the major product. The total number of methyl $\left(-\mathrm{CH}_{3}\right)$

group(s) in compound $\mathbf{S}$ is $\qquad$

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into $\mathrm{Pb}-206$ while the rest of it remains undisintegrated. When the age of the sample is $\mathbf{P} \times 10^8$ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of $\mathbf{P}$ is $\qquad$ ………. [Given: Half-life of U-238 is $4.5 \times 10^9$ years; $\log _{\mathrm{e}} 2=0.693$ ]

Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $\quad \mathrm{R}$ has exactly 6 elements. ii. For each $(a, b) \in R$,we have $|a-b| \geq 2$. Let $Y=\{R \in X$ : The range of $R$ has exactly one element $\}$ and $Z=\{R \in X: R$ is a function from $S$ to $S\}$. Let $n(A)$ denote the number of elements in a set $A$. (There are two questions based on PARAGRAPH "I", the question given below is one of them)

If $n(X)={ }^{m} C_{6}$, then the value of $m$ is $\qquad$

If the value of $n(Y)+n(Z)$ is $k^{2}$, then $|k|$ is $\qquad$

Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]$ be the function defined by $f(x)=\sin ^{2} x$ and let $g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)$ be the function defined by $g=\sqrt{\frac{\pi x}{2}-x^{2}}$.

The value of $2 \int_{0}^{\frac{\pi}{2}} f(x) g(x) d x-\int_{0}^{\frac{\pi}{2}} g(x) d x$ is $\qquad$

Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]$ be the function defined by $f(x)=\sin ^{2} x$ and let $g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)$ be the function defined by $g=\sqrt{\frac{\pi x}{2}-x^{2}}$.

The value of $\frac{16}{\pi^{3}} \int_{0}^{\frac{\pi}{2}} f(x) g(x) d x$ is $\qquad$

In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of 0.8 mm . The distance between the slits at time $t$ is given by $\mathrm{d}=(0.8+0.04 \sin \omega \mathrm{t}) \mathrm{mm}$, where $\omega=0.08 \mathrm{rads}^{-1}$. The distance of the screen from the slits is 1 m and the wavelength of the light used to illuminate the slits is $6000 AA$. The interference pattern on the screen changes with time,while the central bright fringe (zeroth fringe) remains fixed at point O

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

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

Two particles, 1 and 2 , each of mass $m$, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $x_{0}$, are oscillating with amplitude a and angular frequency $\omega$. Thus, their positions at time $t$ are given by $x_{1}(t)=\left(x_{0}+d\right)+a \sin \omega t$ and $x_{2}(t)=\left(x_{0}-d\right)-a \sin \omega t$, respectively, where $d>2 a$. Particle 3 of mass $m$ moves towards this system with speed $u_{0}=a \omega / 2$, and undergoes instantaneous elastic collision with particle 2, at time $t_{0}$. Finally, particles 1 and 2 acquire a center of mass speed $\mathrm{v}_{\mathrm{cm}}$ and oscillate with amplitude b and the same angular frequency $\omega$.

If the collision occurs at time $t_{0}=\pi /(2 \omega)$, then the value of $4 b^{2} / a^{2}$ will be

An organic compound $\mathbf{P}$ with molecular formula $\mathrm{C}_{9} \mathrm{H}_{18} \mathrm{O}_{2}$ decolorizes bromine water and also shows positive iodoform test. $\mathbf{P}$ on ozonolysis followed by treatment with $\mathrm{H}_{2} \mathrm{O}_{2}$ gives $\mathbf{Q}$ and $\mathbf{R}$. While compound $\mathbf{Q}$ shows positive iodoform test, compound $\mathbf{R}$ does not give positive iodoform test. $\mathbf{Q}$ and $\mathbf{R}$ on oxidation with pyridinium chlorochromate (PCC) followed by heating give $\mathbf{S}$ and $\mathbf{T}$, respectively. Both $\mathbf{S}$ and $\mathbf{T}$ show positive iodoform test. Complete copolymerization of 500 moles of $\mathbf{Q}$ and 500 moles of $\mathbf{R}$ gives one mole of a single acyclic copolymer U. [Given, atomic mass: $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16$ ]

The molecular weight of $\mathbf{U}$ is