JEE Advanced 2023 · 04 Jun 2023 · Paper 2

Q1.An electric dipole is formed by two charges $+q$ and $-q$ located in xy-plane at $(0,2)$ mm and $(0,-2)$ mm, respectively, as shown in the figure. The electric potential at point $\mathrm{P}(100,100)$ mm due to the dipole is $V_0$. The charges $+q$ and $-q$ are then moved to th …

Q2.Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck's constant $h$ and the speed of light c , as $Y={\mathrm{c}}^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option ?

Q3.A particle of mass m is moving in the $xy$-plane such that its velocity at a point $(x,y)$ is given as $\vec{v}=\alpha (y\hat{x}+2x\hat{y})$, where $\alpha$ is a non-zero constant. What is the force $\vec{F}$ acting on the particle?

Q4.An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas in $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the c …

Q5.A monochromatic light wave is incident normally on a glass slab of thickness d , as shown in the figure. The refractive index of the slab increases linearly from ${\mathrm{n}}_1$ to ${\mathrm{n}}_2$ over the height h . Which of the following statement(s) is (are) true about the l …

Q6.An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $J_0\hat{x}$ is applied at a height h above the center of the disk. If $h=h_m$ then the d …

Q7.The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}=30(2\hat{x}+\hat{y})\sin \left[2\pi \left(5\times 10^{14}t-\frac{10^7}{3}z\right)\right]\mathrm{V}{\mathrm{m}}^{-1}$. Which of the following option(s) is(are) …

Q8.A thin circular coin of mass 5 gm and radius 4/3 cm is initially in a horizontal $xy$-plane. The coin is tossed vertically up ( $+z$ direction) by applying an impulse of $\sqrt{\frac{\pi }{2}}\times 10^{-2}\mathrm{N}$-s at a distance $2/3\mathrm{cm}$ from its center. The co …

Q9.A rectangular conducting loop of length 4 cm and width 2 cm is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}$ with a constant speed $v$. The wire is carryi …

Q10.A string of length 1 m and mass $2\times 10^{-5}\mathrm{kg}$ is under tension T . when the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz. The value of tension T is ____ Newton.

Q11.An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from …

Q12.In a radioactive decay process, the activity is defined as $A=-\frac{\mathrm{d}N}{\mathrm{d}t}$, where $N(t)$ is the number of radioactive nuclei at time $t$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of …

Q13.One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the $P-V$ diagrams below. In cycle I, processes $a,b,c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes $a^{\prime },b^{\prime },c^{\prime }$ an …

Q14.$S_1$ and $S_2$ are two identical sound sources of frequency 656 Hz. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2}{\mathrm{ms}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three …

Q15.$S_1$ and $S_2$ are two identical sound sources of frequency 656 Hz. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2}{\mathrm{ms}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three …

Q1.The correct molecular orbital diagram for ${\mathrm{F}}_2$ molecule in the ground state is

Q2.Consider the following statements related to colloids.
(I) Lyophobic colloids are not formed by simple mixing of dispersed phase and dispersion medium.
(II) For emulsions, both the dispersed phase and the dispersion medium are liquid.
(III) Micelles are produced by dissolving a s …

Q3.In the following reactions, P, Q, R, and S are the major products.

(see structures for reaction schemes)

The correct statement about $\mathbf{P},\mathbf{Q},\mathbf{R}$, and $\mathbf{S}$ is

Q4.A disaccharide X cannot be oxidised by bromine water. The acid hydrolysis of X leads to a laevorotatory solution. The disaccharide $\mathbf{X}$ is

Q5.The complex(es), which can exhibit the type of isomerism shown by $\left[\mathrm{Pt}{\left({\mathrm{NH}}_3\right)}_2{\mathrm{Br}}_2\right]$, is(are)

Q6.Atoms of metals $\mathrm{x},\mathrm{y}$, and z form face-centred cubic (fcc) unit cell of edge length ${\mathrm{L}}_{\mathrm{x}}$, body-centred cubic (bcc) unit cell of edge length ${\mathrm{L}}_{\mathrm{y}}$, and simple cubic unit cell of edge length …

Q7.In the following reactions, P, Q, R, and S are the major products.

(see structures for reaction schemes)

The correct statement (s) about $\mathbf{P},\mathbf{Q},\mathbf{R}$, and $\mathbf{S}$ is (are)

Q8.${\mathrm{H}}_2\mathrm{S}$ (5 moles) reacts completely with acidified aqueous potassium permanganate solution. In this reaction, the number of moles of water produced is $\mathbf{x}$, and the number of moles of electrons involved is $\mathbf{y}$. The value of …

Q9.Among ${\left[{\mathrm{I}}_3\right]}^{+},{\left[{\mathrm{SiO}}_4\right]}^{4-},{\mathrm{SO}}_2{\mathrm{Cl}}_2,{\mathrm{XeF}}_2,{\mathrm{SF}}_4,{\mathrm{ClF}}_3,\mathrm{Ni}(\mathrm{CO}{)}_4,{\mathrm{XeO}}_2{\mathrm{F}}_2,{\left[{\mathrm{PtCl}}_4\right]}^{2-},{\mathrm{XeF}}_4$ …

Q10.Consider the following molecules : ${\mathrm{Br}}_3{\mathrm{O}}_8,{\mathrm{F}}_2\mathrm{O},{\mathrm{H}}_2{\mathrm{S}}_4{\mathrm{O}}_6,{\mathrm{H}}_2{\mathrm{S}}_5{\mathrm{O}}_6$, and ${\mathrm{C}}_3{\mathrm{O}}_2$.
Count the number of atoms existing in their zero …

Q11.For ${\mathrm{He}}^{+}$, a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm.
The wavelength (in nm) of the emitted photon during the transition is ____.
[Use:
Bohr radius, $a=52.9$ pm
Rydberg constant, …

Q12.50 mL of 0.2 molal urea solution (density $=1.012\mathrm{g}{\mathrm{mL}}^{-1}$ at 300 K ) is mixed with 250 mL of a solution containing 0.06 g of urea. Both the solutions were prepared in the same solvent. The osmotic pressure (in Torr) of the resulting solution at 300 K is _ …

Q13.The reaction of 4-methyloct-ene (P, 2.52 g) with HBr in the presence of ${\left({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{CO}\right)}_2{\mathrm{O}}_2$ gives two isomeric bromides in a 9 : 1 ratio, with combined yield of $50\%$. Of these, the entire amount of the primary alkyl b …

Q14.The entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given.
${\mathrm{S}}_{\mathrm{T}}$ and ${\mathrm{S}}_0$ are entropies of the phases at temperatures T and 0 K, respectively.

(see diagram)

The transition temperature for $\alpha$ to …

Q15.The entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given.
${\mathrm{S}}_{\mathrm{T}}$ and ${\mathrm{S}}_0$ are entropies of the phases at temperatures T and 0 K, respectively.

(see diagram)

The transition temperature for $\alpha$ to …

Q16.A trinitro compound, 1, 3,5 tris-(4-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0^{\circ }\mathrm{C}$ provides P as the product. P, …

Q17.A trinitro compound, 1, 3,5 tris-(4-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0^{\circ }\mathrm{C}$ provides P as the product. P, …

Q1.Let $\mathrm{f}:[1,\infty )\to \mathbb{R}$ be a differentiable function such that $\mathrm{f}(1)=\frac{1}{3}$ and $3{\int }_1^{\mathrm{x}}\mathrm{f}(\mathrm{t})\mathrm{dt}=\mathrm{xf}(\mathrm{x})-\frac{{\mathrm{x}}^3}{3},\mathrm{x}\in [1,\infty )$. Let $e$ denote the …

Q2.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

Q3.For any $\mathrm{y}\in \mathbb{R}$, let ${\cot }^{-1}(\mathrm{y})\in (0,\pi )$ and ${\tan }^{-1}(\mathrm{y})\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation …

Q4.Let the position vectors of the points $P,Q,R$ and $S$ be $\vec{a}=\hat{i}+2\hat{j}-5\hat{k},\vec{b}=3\hat{i}+6\hat{j}+3\hat{k}$, $\overrightarrow{\mathrm{c}}=\frac{17}{5}\hat{\mathrm{i}}+\frac{16}{5}\hat{\mathrm{j}}+7\hat{\mathrm{k}}$ and …

Q5.Let $\mathrm{M}=({\mathrm{a}}_{\mathrm{ij}}),\mathrm{i},\mathrm{j}\in \{1,2,3\}$, be the $3\times 3$ matrix such that ${\mathrm{a}}_{\mathrm{ij}}=1$ if $\mathrm{j}+1$ is divisible by i , otherwise ${\mathrm{a}}_{\mathrm{ij}}=0$. Then which of the following statements is (are) true ?

Q6.Let $\mathrm{f}:(0,1)\to \mathbb{R}$ be the function defined as $\mathrm{f}(\mathrm{x})=[4\mathrm{x}]{\left(\mathrm{x}-\frac{1}{4}\right)}^2\left(\mathrm{x}-\frac{1}{2}\right)$, where [x] denotes the greatest integer less than or equal to x . Then which of the following …

Q7.Let S be the set of all twice differentiable functions f from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{{\mathrm{d}}^2\mathrm{f}}{{\mathrm{dx}}^2}(\mathrm{x})>0$ for all $\mathrm{x}\in (-1,1)$. For $\mathrm{f}\in \mathrm{S}$, let ${\mathrm{X}}_{\mathrm{f}}$ be the number of p …

Q8.For $\mathrm{x}\in \mathbb{R}$, let ${\tan }^{-1}(\mathrm{x})\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $\mathrm{f}:\mathbb{R}\to \mathbb{R}$ defined by …

Q9.For $\mathrm{x}\in \mathbb{R}$, let $\mathrm{y}(\mathrm{x})$ be a solution of the differential equation $\left({\mathrm{x}}^2-5\right)\frac{\mathrm{dy}}{\mathrm{dx}}-2\mathrm{xy}=-2\mathrm{x}{\left({\mathrm{x}}^2-5\right)}^2$ such that $\mathrm{y}(2)=7$.
Then the maximum valu …

Q10.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 mu …

Q11.Let ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3,\dots ,{\mathrm{A}}_8$ be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let ${\mathrm{PA}}_{\mathrm{i}}$ denote the distance between the points P and …

Q12.Let $\mathrm{R}=\left\{\left(\begin{array}{lll}\mathrm{a}& 3& \mathrm{b}\\ \mathrm{c}& 2& \mathrm{d}\\ 0& 5& 0\end{array}\right):\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\in \{0,3,5,7,11,13,17,19\}\right\}$. Then the number of invertible matrices in R is

Q13.Let ${\mathrm{C}}_1$ be the circle of radius 1 with center at the origin. Let ${\mathrm{C}}_2$ be the circle of radius r with center at the point $\mathrm{A}=(4,1)$, where $1<\mathrm{r}<3$. Two distinct common tangents PQ and ST of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are drawn. …

Q14.Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

Let a be the area of the triangle …

Q15.Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.

Then the inradius of the triangle …

Q16.Consider the $6\times 6$ square in the figure. Let ${\mathrm{A}}_1,{\mathrm{A}}_2,\dots ,{\mathrm{A}}_{49}$ be the points of intersections (dots in the picture) in some order. We say that ${\mathrm{A}}_i$ and ${\mathrm{A}}_j$ are friends if they are adjacent along a row …

Q17.Consider the $6\times 6$ square in the figure. Let ${\mathrm{A}}_1,{\mathrm{A}}_2,\dots ,{\mathrm{A}}_{49}$ be the points of intersections (dots in the picture) in some order. We say that ${\mathrm{A}}_i$ and ${\mathrm{A}}_j$ are friends if they are adjacent along a row …