JEE Advanced 2024 · 26 May 2024 · Paper 2

Q1.A region in the form of an equilateral triangle (in x-y plane) of height $L$ has a uniform magnetic field $\overrightarrow{\mathrm{B}}$ 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 …

Q2.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 ${\mathrm{r}}_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central forc …

Q3.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 m …

Q4.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 v …

Q5.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 orien …

Q6.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 $d=0.7\mathrm{m}$ below the water surface and then released from rest. It emerges from the water surf …

Q7.A positive, singly ionized atom of mass number $A_{\mathrm{M}}$ 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 t …

Q8.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 ____.

Q9.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 …

Q10.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 elect …

Q11.Two equilateral-triangular prisms ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism ${\mathrm{P}}_1$ at an angle of incidence $\theta$ such that the outgoing ray undergoes minimum de …

Q12.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 …

Q13.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 $\Delta P=144\mathrm{Pa}$. Now, the chamber pressure is reduced to $8P_0/27$ so that the bubble radius and its excess …

Q14.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 $d=(0.8+0.04\sin \omega t)\mathrm{mm}$, where …

Q15.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 $d=(0.8+0.04\sin \omega t)\mathrm{mm}$, where …

Q16.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 $\omega$. …

Q17.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 $\omega$. …

Q1.According to Bohr's model, the highest kinetic energy is associated with the electron in the :

Q2.In a metal deficient oxide sample, ${\mathbf{M}}_{\mathbf{x}}{\mathbf{Y}}_2{\mathbf{O}}_4$ ($\mathbf{M}$ and $\mathbf{Y}$ are metals), $\mathbf{M}$ is present in both +2 and +3 oxidation states and $\mathbf{Y}$ is in +3 oxidation state. If the fraction …

Q3.In the following reaction sequences, the major product $\mathbf{Q}$ is:

Q4.The species formed on fluorination of phosphorus pentachloride in a polar organic solvent are :

Q5.An aqueous solution of hydrazine $\left({\mathrm{N}}_2{\mathrm{H}}_4\right)$ is electrochemically oxidized by ${\mathrm{O}}_2$, thereby releasing chemical energy in the form of electrical energy. One of the products generated from the electrochemical reaction is …

Q6.The option(s) with correct sequence of reagents for the conversion of $\mathbf{P}$ to $\mathbf{Q}$ is(are) :

Q7.The compound(s) having peroxide linkage is(are) :

Q8.To form a complete monolayer of acetic acid on 1g 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 …

Q9.Vessel-1 contains ${\mathbf{w}}_2\mathbf{g}$ of a non-volatile solute $\mathbf{X}$ dissolved in ${\mathbf{w}}_{1}g$ of water. Vessel-2 contains ${\mathbf{w}}_{2}g$ of another non-volatile solute $\mathbf{Y}$ dissolved in …

Q10.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- …

Q11.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 $\mathbf{P}\times 10^8$ years, the ratio of mass of $\mathrm{Pb}-206$ to tha …

Q12.Among ${\left[\mathrm{Co}(\mathrm{CN}{)}_4\right]}^{4-},\left[\mathrm{Co}(\mathrm{CO}{)}_3(\mathrm{NO})\right],{\mathrm{XeF}}_4,{\left[{\mathrm{PCl}}_4\right]}^{+},{\left[{\mathrm{PdCl}}_4\right]}^{2-},{\left[{\mathrm{ICl}}_4\right]}^{-},{\left[\mathrm{Cu}(\mathrm{CN}{)}_4\right]}^{3-}$ …

Q13.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 P reacts with 3 equivalents of ${\mathrm{NH}}_2\mathrm{OH}$ to produce oxime $\mathbf{Q}$. …

Q14.An organic compound P with molecular formula ${\mathrm{C}}_9{\mathrm{H}}_{18}{\mathrm{O}}_2$ decolorizes bromine water and also shows positive iodoform test. P on ozonolysis followed by treatment with ${\mathrm{H}}_2{\mathrm{O}}_2$ gives Q and R. While compound Q shows posit …

Q15.An organic compound P with molecular formula ${\mathrm{C}}_9{\mathrm{H}}_{18}{\mathrm{O}}_2$ decolorizes bromine water and also shows positive iodoform test. P on ozonolysis followed by treatment with ${\mathrm{H}}_2{\mathrm{O}}_2$ gives Q and R. While compound Q shows posit …

Q16.When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex P is formed. In a strong acidic medium, the equilibrium shifts completely towards $\mathbf{P}$. Addition of zinc chloride to $\mathbf{P}$ in a …

Q17.When potassium iodide is added to an aqueous solution of potassium ferricyanide, a reversible reaction is observed in which a complex P is formed. In a strong acidic medium, the equilibrium shifts completely towards P. Addition of zinc chloride to P in a slightly acidic medium re …

Q1.Considering only the principal values of the inverse trigonometric functions, the value of

is

Q2.Let $S=\{(x,y)\in \mathbb{R}\times \mathbb{R}:x\ge 0,y\ge 0,y^2\le 4x,y^2\le 12-2x$ and $3y+\sqrt{8}x\le 5\sqrt{8}\}$. If the area of the region $S$ is $\alpha \sqrt{2}$, then $\alpha$ is equal to

Q3.Let $k\in \mathbb{R}$. If ${\lim }_{x\to 0^{+}}(\sin (\sin kx)+\cos x+x{)}^{\frac{2}{x}}=e^6$, then the value of $k$ is

Q4.Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by

Then which of the following statements is TRUE ?

Q5.Let $S$ be the set of all $(\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}$ such that

Then w …

Q6.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 plan …

Q7.Let $A_1,B_1,C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1{C}_1$ and $B_1{C}_1$ 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?

Q8.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 …

Q9.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 $B_i$ deno …

Q10.Let the function $f:\mathbb{R}\to \mathbb{R}$ be defined by

Then the number of solut …

Q11.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 …

Q12.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 $A$ …

Q13.Let the function $f:[1,\infty )\to \mathbb{R}$ be defined by …

Q14.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. $R$ has exactly 6 elements.
ii. For each $(a,b)\in R$, we have $|a-b|\ge 2$.
Let $Y=\{R\in X$ : The range of $R$ has exactly one element $\}$ and …

Q15.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. R has exactly 6 elements.
ii. For each $(a,b)\in R$, we have $|a-b|\ge 2$.

Let $Y=\{R\in X$ : The range of R has exactly one element $\}$ and …

Q16.Let $f:\left[0,\frac{\pi }{2}\right]\to [0,1]$ be the function defined by $f(x)={\sin }^{2}x$ and let $g:\left[0,\frac{\pi }{2}\right]\to [0,\infty )$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$.

The value of …

Q17.Let $f:\left[0,\frac{\pi }{2}\right]\to [0,1]$ be the function defined by $f(x)={\sin }^{2}x$ and let $g:\left[0,\frac{\pi }{2}\right]\to [0,\infty )$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$.

The value of …