JEE Advanced 2022 · 28 Aug 2022 · Paper 2

Q1.A particle of mass 1 kg is subjected to a force which depends on the position as $\vec{F}=-k(x\hat{i}+y\hat{j})kgm{s}^{-2}$ with $k=1kg{s}^{-2}$. At time $t=0$, the particle's position $\vec{r}=\left(\frac{1}{\sqrt{2}}\hat{i}+\sqrt{2}\hat{j}\right)m$ and its velocit …

Q2.In a radioactive decay chain reaction, ${}_{90}^{230}\mathrm{Th}$ nucleus decays into ${}_{84}^{214}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of ${\beta }^{-}$particles emitted in this process is ____.

Q3.Two resistances $R_1=X\Omega$ and $R_2=1\Omega$ are connected to a wire $AB$ of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from $0.2mm$ at $A$ to $1mm$ at $B$. A galvanometer (G) connected to the center …

Q4.In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {ϵ}_0}$, where ${ϵ}_0$ is the permittivity of vacuum. In terms …

Q5.Consider a configuration of $n$ identical units, each consisting of three layers. The first layer is a column of air of height $h=\frac{1}{3}cm$, and the second and third layers are of equal thickness $d=\frac{\sqrt{3}-1}{2}cm$, and refractive indices …

Q6.A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is $\frac{nq}{6{ϵ}_0}$ (in SI units). The value …

Q7.On a frictionless horizontal plane, a bob of mass $m=0.1kg$ is attached to a spring with natural length $l_0=0.1m$. The spring constant is $k_1=0.009N{m}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016N{m}^{-1}$ when $l<l_0$. Initially the bob …

Q8.An object and a concave mirror of focal length $f=10cm$ both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_0=15cm{s}^{-1}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirr …

Q9.In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as ${\rho }_A=kr$, where $k$ is positive. In the spherical shell $B$ of outer radius …

Q10.In Circuit-1 and Circuit-2 shown in the figures, $R_1=1\Omega ,R_2=2\Omega$ and $R_3=3\Omega$. $P_1$ and $P_2$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $S_1$ and $S_2$ are in open conditions, respectively.
$Q_1$ and …

Q11.A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma =\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a1}$, the radius of the bubble is fou …

Q12.A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is

A particle of positive charg …

Q13.A double slit setup is shown in the figure. One of the slits is in medium 2 of refractive index $n_2$. The other slit is at the interface of this medium with another medium 1 of refractive index $n_1(\ne n_2)$. The line joining the slits is perpendicular to the interfa …

Q14.In the given $P-V$ diagram, a monoatomic gas $\left(\gamma =\frac{5}{3}\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$. [Given: …

Q15.A flat surface of a thin uniform disk $A$ of radius $R$ is glued to a horizontal table. Another thin uniform disk $B$ of mass $M$ and with the same radius $R$ rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies …

Q16.When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0V$. This potential drops to $0.6V$ if another source with wavelength four times that of the first one and intensity half of the first o …

Q17.Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is 0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

| Mea …

Q18.Which one of the following options represents the magnetic field $\vec{B}$ at O due to the current flowing in the given wire segments lying on the $xy$ plane?

Q1.Concentration of ${\mathrm{H}}_2{\mathrm{SO}}_4$ and ${\mathrm{Na}}_2{\mathrm{SO}}_4$ in a solution is 1 M and $1.8\times 10^{-2}\mathrm{M}$, respectively. Molar solubility of ${\mathrm{PbSO}}_4$ in the same solution is …

Q2.An aqueous solution is prepared by dissolving 0.1 mol of an ionic salt in 1.8 kg of water at 35 °C. The salt remains 90% dissociated in the solution. The vapour pressure of the solution is 59.724 mm of Hg. Vapor pressure of water at 35 °C is 60.000 mm of Hg. The number of ions pr …

Q3.Consider the strong electrolytes ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$, ${\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$. Limiting molar conductivity ( ${\Lambda }^o$ ) of …

Q4.The reaction of Xe and ${\mathrm{O}}_2{\mathrm{F}}_2$ gives a Xe compound P. The number of moles of HF produced by the complete hydrolysis of 1 mol of $\mathbf{P}$ is ____.

Q5.Thermal decomposition of ${\mathrm{AgNO}}_3$ produces two paramagnetic gases. The total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons is ____.

Q6.The number of isomeric tetraenes (NOT containing $sp$-hybridized carbon atoms) that can be formed from the following reaction sequence is ____.

Q7.The number of -${\mathrm{CH}}_2$- (methylene) groups in the product formed from the following reaction sequence is ____.

Q8.The total number of chiral molecules formed from one molecule of P on complete ozonolysis ( ${\mathrm{O}}_3,\mathrm{Zn}/{\mathrm{H}}_2\mathrm{O}$ ) is ____.

Q9.To check the principle of multiple proportions, a series of pure binary compounds $\left({\mathrm{P}}_{\mathrm{m}}{\mathrm{Q}}_{\mathrm{n}}\right)$ were analyzed and their composition is tabulated below. The correct option(s) is(are)
| Compound | Weight % of P | Weight % of Q || …

Q10.The correct option(s) about entropy (S) is(are)
[R = gas constant, F = Faraday constant, T = Temperature]

Q11.The compound(s) which react(s) with ${\mathrm{NH}}_3$ to give boron nitride (BN) is(are)

Q12.The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500\mathrm{K}$ is(are)

Q13.Considering the following reaction sequence, the correct statement(s) is(are)

Q14.Among the following, the correct statement(s) about polymers is(are)

Q15.Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to

Q16.The reaction of ${\mathrm{HClO}}_3$ with HCl gives a paramagnetic gas, which upon reaction with ${\mathrm{O}}_3$ produces

Q17.The reaction $\mathrm{Pb}{\left({\mathrm{NO}}_3\right)}_2$ and NaCl in water produces a precipitate that dissolves upon the addition of HCl of appropriate concentration. The dissolution of the precipitate is due to the formation of

Q18.Treatment of D-glucose with aqueous NaOH results in a mixture of monosaccharides, which are

Q1.Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi }{4}<\beta <0<\alpha <\frac{\pi }{4}$. If $\sin (\alpha +\beta )=\frac{1}{3}$ and $\cos (\alpha -\beta )=\frac{2}{3}$, then the greatest integer less than or equal to …

Q2.If $y(x)$ is the solution of the differential equation

and the slope of the curve $y=y(x)$ is never zero, then the value of $10y(\sqrt{2})$ is ____.

Q3.The greatest integer less than or equal to

is ____.

Q4.The product of all positive real values of x satisfying the equation

is ____.

Q5.If

then the value of $6\beta$ is ____.

Q6.Let $\beta$ be a real number. Consider the matrix

If $A^7-(\beta -1)A^6-\beta A^5$ is a singular matrix, then the value of $9\beta$ is ____.

Q7.Consider the hyperbola

with foci at S and $S_1$, where S lies on the positive x-axis. Let P be a point on the hyperbola, in the first quadrant. Let $\angle SP{S}_1=\alpha$, with $\alpha <\frac{\pi }{2}$. The straight line passing t …

Q8.Consider the functions $f,g:\mathbb{R}\to \mathbb{R}$ defined by

If $\alpha$ is the area of the region …

Q9.Let PQRS be a quadrilateral in a plane, where $QR=1$, $\angle PQR=\angle QRS=70^{\circ }$, $\angle PQS=15^{\circ }$ and $\angle PRS=40^{\circ }$. If $\angle RPS={\theta }^{\circ }$, $PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of …

Q10.Let

Let $g:[0,1]\to \mathbb{R}$ be the function defined by

Then, which of the following statements is/are TRUE?

Q11.Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of

are integers, then which of the following is/are possible value(s) of $|z|$?

Q12.Let G be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be n circles of equal radius $r>0$. Suppose each of the n circles $G_1,G_2,\dots ,G_n$ touches the circle G externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ extern …

Q13.Let $\hat{i},\hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let …

Q14.For $x\in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation

Then, which of the following statements is/are TRUE?

Q15.Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

Q16.If $M=\left(\begin{array}{cc}\frac{5}{2}& \frac{3}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$?

Q17.Suppose that
Box-I contains 8 red, 3 blue and 5 green balls,
Box-II contains 24 red, 9 blue and 15 green balls,
Box-III contains 1 blue, 12 green and 3 yellow balls,
Box-IV contains 10 green, 16 orange and 6 white balls.

A ball is chosen randomly from Box-I; call this ball $b$ …

Q18.For positive integer n, define

Then, the value of ${\lim }_{n\to \infty }f(n)$ is equal to