JEE Advanced 2018 — Paper 2

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x axis. Its kinetic energy K changes with time as $\mathrm{dK} / \mathrm{dt}=\gamma \mathrm{t}$, where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_{0}$. Which of the following statements is (are) true?

An infinitely long thin non-conducting wire is parallel to the z -axis and carries a uniform line charge density $\lambda$. It pierces a thin nonconducting spherical shell of radius $R$ in such a way that the arc $P Q$ subtends an angle $120^{\circ}$ at the centre $O$ of the spherical shell, as shown in the figure. The permittivity of free space is $\varepsilon_{0}$. Which of the following statements is (are) true?

In a radioactive decay chain, ${ }_{90}^{232} \mathrm{Th}$ nucleus decays to ${ }_{82}^{212} \mathrm{~Pb}$ nucleus. Let $N_{\alpha}$ and $N_{\beta}$ be the number of $\alpha$ and $\beta^{-}$particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50.7 cm and 83.9 cm . Which of the following statements is (are) true?

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4 \mathrm{~kg}$ is at rest on this surface. An impulse of 1.0 Ns is applied to the block at time $t=0$ so that it starts moving along the x -axis with a velocity $\mathrm{v}(\mathrm{t})=\mathrm{v}_{0} \mathrm{e}^{-\mathrm{t} / \tau}$, where $\mathrm{v}_{0}$ is a constant and $\tau=4 \mathrm{~s}$. The displacement of the block, in meters, at $\mathrm{t}=\tau$ is $\_\_\_\_$ Take $\mathrm{e}^{-1}=0.37$.

A ball is projected from the ground at an angle of $45^{\circ}$ with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30^{\circ}$ with the horizontal surface. The maximum height it reaches after the bounce, in metres, is $\_\_\_\_$ -.

A particle, of mass $10^{-3} \mathrm{~kg}$ and charge 1.0 C , is initially at rest. At time $\mathrm{t}=0$, the particle comes under the influence of an electric field $\overrightarrow{\mathrm{E}}(\mathrm{t})=\mathrm{E}_{0} \sin \omega \mathrm{t} \hat{\mathrm{i}}$, where $\mathrm{E}_{0}=1.0 \mathrm{NC}^{-1}$ and $\omega=10^{3} \mathrm{rad} \mathrm{s}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $\mathrm{m} \mathrm{s}^{-1}$, attained by the particle at subsequent times is $\_\_\_\_$ .

A moving coil galvanometer has 50 turns and each turn has an area $2 \times 10^{-4} \mathrm{~m}^{2}$. The magnetic field produced by the magnet inside the galvanometer is 0.02 T . The torsional constant of the suspension wire is $10^{-4} \mathrm{~N} \mathrm{~m} \mathrm{rad}^{-1}$. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by 0.2 rad . The resistance of the coil of the galvanometer is $50 \Omega$. This galvanometer is to be converted into an ammeter capable of measuring current in the range $0-1.0 \mathrm{~A}$. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is $\_\_\_\_$.

A steel wire of diameter 0.5 mm and Young's modulus $2 \times 10^{11} \mathrm{Nm}^{-2}$ carries a load of mass M. The length of the wire with the load is 1.0 m . A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 kg , the vernier scale division which coincides with a main scale division is $\_\_\_\_$. Take $\mathrm{g}=10 \mathrm{~ms}^{-2}$ and is $\pi=3.2$.

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant $\mathrm{R}=8.0 \mathrm{~J}$ $\mathrm{mol}^{-1} \mathrm{~K}^{-1}$, the decrease in its internal energy, in Joule, is $\_\_\_\_$.

In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 eV . The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is $100 \%$. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $\mathrm{F}=\mathrm{n} \times 10^{-4} \mathrm{~N}$ due to the impact of the electrons. The value of n is $\_\_\_\_$ .

Mass of the electron $\mathrm{m}_{\mathrm{e}}=9 \times 10^{-31} \mathrm{~kg}$ and $1.0 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}$.

Consider a hydrogen-like ionized atom with atomic number Z with a single electron. In the emission spectrum of this atom, the photon emitted in the $\mathrm{n}=2$ to $\mathrm{n}=1$ transition has energy 74.8 eV higher than the photon emitted in the $\mathrm{n}=3$ to $\mathrm{n}=2$ transition. The ionization energy of the hydrogen atom is 13.6 eV . The value of Z is $\_\_\_\_$ .

The electric field E is measured at a point $\mathrm{P}(0,0, \mathrm{~d})$ generated due to various charge distributions and the dependence of E on d is found to be different for different charge distributions. List-I contains different relations between $E$ and d. List-II describes different electric charge distributions, along with their locations. Match the functions in List-I with the related charge distributions in List-II.

A planet of mass $M$, has two natural satellites with masses $m_{1}$ and $m_{2}$. The radii of their circular orbits are $R_{1}$ and $R_{2}$ respectively. Ignore the gravitational force between the satellites. Define $v_{1}, L_{1}, K_{1}$ and $T_{1}$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1 ; and $v_{2}, L_{2}, K_{2}$ and $T_{2}$ to be the corresponding quantities of satellite 2 . Given $m_{1} / m_{2}=2$ and $R_{1} / R_{2}=1 / 4$, match the ratios in List-I to the numbers in List-II.

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PVdiagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{\mathrm{p}}$ is the linear momentum, $\overrightarrow{\mathrm{L}}$ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

The correct option(s) regarding the complex $\left[\mathrm{Co}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{3+}$

$\left(\mathrm{en}=\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}\right)$ is (are)

The correct option(s) to distinguish nitrate salts of $\mathrm{Mn}^{2+}$ and $\mathrm{Cu}^{2+}$ taken separately is (are)

For a first order reaction $\mathrm{A}(\mathrm{g}) \rightarrow 2 \mathrm{~B}(\mathrm{~g})+\mathrm{C}(\mathrm{g})$ at constant volume and 300 K , the total pressure at the beginning $(t=0)$ and at time $t$ are $P_{0}$ and $P_{t}$, respectively. Initially, only $A$ is present with concentration $[A]_{0}$, and $t_{1 / 3}$ is the time required for the partial pressure of $A$ to reach $1 / 3^{\text {rd }}$ of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)

For a reaction, $\mathrm{A} \rightleftharpoons \mathrm{P}$, the plots of $[\mathrm{A}]$ and $[\mathrm{P}]$ with time at temperatures $T_{1}$ and $T_{2}$ are given below.

If T$_2$ > T$_1$, the correct statement(s) is (are) (Assume $\Delta$H$^0$ and $\Delta$S$^0$ are independent of temperature and ratio of $\ln K$ at $\mathrm{T}_{1}$ to $\ln K$ at $\mathrm{T}_{2}$ is greater than $\mathrm{T}_{2} / \mathrm{T}_{1}$. Here $H, S, G$ and $K$ are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)

Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of

air is stopped, but the heating is continued in a closed furnace such that the contents undergo self-reduction.

The weight (in kg ) of Pb produced per kg of $\mathrm{O}_{2}$ consumed is $\_\_\_\_$ .

(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~Pb}=207$ )

To measure the quantity of $\mathrm{MnCl}_{2}$ dissolved in an aqueous solution, it was completely converted to $\mathrm{KMnO}_{4}$

using the reaction, $\mathrm{MnCl}_{2}+\mathrm{K}_{2} \mathrm{~S}_{2} \mathrm{O}_{8}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{KMnO}_{4}+\mathrm{H}_{2} \mathrm{SO}_{4}+\mathrm{HCl}$ (equation not

balanced). Few drops of concentrated HCl were added to this solution and gently warmed. Further, oxalic acid

( 225 mg ) was added in portions till the colour of the permanganate ion disappeared. The quantity of $\mathrm{MnCl}_{2}$ (in

mg ) present in the initial solution is $\_\_\_\_$

(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{Mn}=55, \mathrm{Cl}=35.5$ )

The surface of copper gets tarnished by the formation of copper oxide. $\mathrm{N}_{2}$ gas was passed to prevent the oxide

formation during heating of copper at 1250 K . However, the $\mathrm{N}_{2}$ gas contains $1 \mathrm{~mole} \%$ of water vapour as

impurity. The water vapour oxidises copper as per the reaction given below:

$2 \mathrm{Cu}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightarrow \mathrm{Cu}_{2} \mathrm{O}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g})$

$\mathrm{p}_{\mathrm{H}_{2}}$ is the minimum partial pressure

of $\mathrm{H}_{2}$ (in bar) needed to prevent the oxidation at 1250 K . The value of $\ln \left(\mathrm{p}_{\mathrm{H}_{2}}\right)$ is $\_\_\_\_$

(Given: total pressure $=1$ bar, $R$ (universal gas constant) $=8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, \ln (10)=2.3 . \mathrm{Cu}(\mathrm{s})$

and $\mathrm{Cu}_{2} \mathrm{O}(\mathrm{s})$ are mutually immiscible.

Consider an electrochemical cell: $\mathrm{A}(\mathrm{s})\left|\mathrm{A}^{\mathrm{n}+}(\mathrm{aq}, 2 \mathrm{M}) \| \mathrm{B}^{2 \mathrm{n}+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{B}(\mathrm{s})$. The value of $\Delta \mathrm{H}^{\ominus}$ for the

cell reaction is twice that of $\Delta \mathrm{G}^{\ominus}$ at 300 K . If the emf of the cell is zero, the $\Delta \mathrm{S}^{\ominus}$ (in $\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ) of the

cell reaction per mole of B formed at 300 K is $\_\_\_\_$ .

(Given: $\ln (2)=0.7$, R (universal gas constant) $=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} . H, S$ and $G$ are enthalpy, entropy

and Gibbs energy, respectively.)

Match each set of hybrid orbitals from LIST-I with complex(es) given in LIST-II. The correct option is

Dilution processes of different aqueous solutions, with water, are given in LIST-I. The effects of dilution of the solutions on $\left[\mathrm{H}^{+}\right]$are given in LIST-II.

(Note: Degree of dissociation ( $\alpha$ ) of weak acid and weak base is $\ll 1$; degree of hydrolysis of salt $\ll 1$; $\left[\mathrm{H}^{+}\right]$represents the concentration of $\mathrm{H}^{+}$ions)

Match each process given in LIST-I with one or more effect(s) in LIST-II. The correct option is

For any positive integer $n$, define $f_{n}:(0, \infty) \rightarrow R$ as $f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)$ (Here, the inverse trigonometric function $\tan ^{-1} \mathrm{x}$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ ) Then, which of the following statement(s) is (are) TRUE ?

Let T be the line passing through the points $\mathrm{P}(-2,7)$ and $\mathrm{Q}(2,-5)$. Let $\mathrm{F}_{1}$ be the set of all pairs of circles $\left(S_{1}, S_{2}\right)$ such that $T$ is tangent to $S_{1}$ at $P$ and tangent to $S_{2}$ at $Q$, and also such that $S_{1}$ and $S_{2}$ touch each other at a point, say, $M$. Let $E_{1}$ be the set representing the locus of $M$ as the pair $\left(S_{1}, S_{2}\right)$ varies in $F_{1}$. Let the set of all straight line segments joining a pair of distinct points of $E_{1}$ and passing through the point $R(1,1)$ be $F_{2}$. Let $E_{2}$ be the set of the mid-points of the line segments in the set $F_{2}$. Then, which of the following statement(s) is (are) TRUE?

Let S be the set of all column matrices $\left[\begin{array}{l}b_{1}\\ b_{2}\\ b_{3}\end{array}\right]$ such that $b_{1}, b_{2}, b_{3} \in \mathrm{R}$ and the system of equations (in real variables) $\begin{aligned} & -x+2 y+5 z=b_{1} \\& 2 x-4 y+3 z=b_{2} \\& x-2 y+2 z=b_{3} \end{aligned}$ has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[\begin{array}{l}b_{1}\\ b_{2}\\ b_{3}\end{array}\right] \in \mathrm{S}$ ?

Consider two straight lines, each of which is tangent to both the circle $x^{2}+y^{2}=\frac{1}{2}$ and the parabola $y^{2}=4 x$. Let these lines intersect at the point Q . Consider the ellipse whose center is at the origin $\mathrm{O}(0,0)$ and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

Let $s, t, r$ be non-zero complex numbers and L be the set of solutions $z=x+i y(x, y \in \mathrm{R}, i=\sqrt{-1})$ of

the equation $s z+t \bar{z}+r=0$, where $\bar{z}=x-i y$. Then, which of the following statement(s) is (are) TRUE?

Let $f:(0, \pi) \rightarrow \mathrm{R}$ be a twice differentiable function such that $\lim _{t \rightarrow x} \frac{f(x) \sin t-f(t) \sin x}{t-x}=\sin ^{2} x \text { for all } x \in(0, \pi)$ If $f\left(\frac{\pi}{6}\right)=-\frac{\pi}{12}$, then which of the following statement(s) is (are) TRUE ?

The value of the integral $\int_{0}^{1 / 2} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{1 / 4}} d x$ is $\_\_\_\_\_$ .

Let P be a matrix of order $3 \times 3$ such that all the entries in P are from the set $\{-1,0,1\}$. Then, the maximum possible value of the determinant of $P$ is $\_\_\_\_\_$ .

Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If $\alpha$ is the number of oneone functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac{1}{5!}(\beta-\alpha)$ is $\_\_\_\_\_\_$ .

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation $\frac{d y}{d x}=(2+5 y)(5 y-2),$ then the value of $\lim _{x \rightarrow-\infty} f(x)$ is $\qquad$ .

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function with $f(0)=1$ and satisfying the equation

$\mathrm{f}(x+y)=\mathrm{f}(x) \mathrm{f}^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathrm{R}$.

Then, the value of $\log _{\mathrm{e}}(f(4))$ is $\qquad$ $-$

Let P be a point in the first octant, whose image Q in the plane $\mathrm{x}+\mathrm{y}=3$ (that is, the line segment PQ is perpendicular to the plane $x+y=3$ and the mid-point of PQ lies in the plane $x+y=3$ ) lies on the $z$-axis. Let the distance of P from the x -axis be 5 . If R is the image of P in the xy-plane, then the length of PR is $\_\_\_\_\_$ .

Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the x -axis, y -axis and z-axis,

respectively, where $\mathrm{O}(0,0,0)$ is the origin. Let $S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ be the centre of the cube and T be the vertex of

the cube opposite to the origin O such that S lies on the diagonal OT. If $\vec{p}=\overrightarrow{S P}, \vec{q}=\overrightarrow{S Q}, \vec{r}=\overrightarrow{S R}$ and

$\vec{t}=\overrightarrow{S T}$, then the value of $|(\vec{p} \times \vec{q}) \times(\vec{r} \times \vec{t})|$ is $\_\_\_\_\_$ .

Let $\mathrm{X}=\left({ }^{10} \mathrm{C}_{1}\right)^{2}+2\left({ }^{10} \mathrm{C}_{2}\right)^{2}+3\left({ }^{10} \mathrm{C}_{3}\right)^{2}+\ldots . .+10\left({ }^{10} \mathrm{C}_{10}\right)^{2}$ where ${ }^{10} \mathrm{C}_{\mathrm{r}}, \mathrm{r} \in\{1,2, \ldots . ., 10\}$ denote binomial coefficients. Then the value of $\frac{1}{1430} \mathrm{X}$ is $\_\_\_\_$ .

Let $E_{1}=\left\{x \in \mathrm{R}: x \neq 1\right.$ and $\left.\frac{x}{x-1}>0\right\}$ and $E_{2}=\left\{x \in \mathrm{E}_{1}: \sin ^{-1}\left(\log _{\mathrm{e}}\left(\frac{x}{x-1}\right)\right)\right.$ is a real number $\}$ (Here, the inverse trigonometric function $\sin ^{-1} \mathrm{x}$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.) Let $\quad f: E_{1} \rightarrow \mathrm{R}$ be the function defined by $f(x)=\log _{e}\left(\frac{x}{x-1}\right)$ and $\quad g: E_{2} \rightarrow \mathrm{R}$ be the function defined by $g(x)=\sin ^{-1}\left(\log _{e}\left(\frac{x}{x-1}\right)\right)$.

In a high school, a committee has to be formed from a group of 6 boys $M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}$ and 5

girls $\mathrm{G}_{1}, \mathrm{G}_{2}, \mathrm{G}_{3}, \mathrm{G}_{4}, \mathrm{G}_{5}$.

(i) Let $\alpha_{1}$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let $\alpha_{2}$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

(iii) Let $\alpha_{3}$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let $\alpha_{4}$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having atleast 2 girls and such that both $M_{1}$ and $G_{1}$ are NOT in the committee together.

Let $\mathrm{H}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, where $a>b>0$, be a hyperbola in the xy-plane whose conjugate axis LM subtends an angle of $60^{\circ}$ at one of its vertices $N$. Let the area of the triangle LMN be $4 \sqrt{3}$.

Let $f_{1}: \mathrm{R} \rightarrow \mathrm{R}, f_{2}:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathrm{R}, \mathrm{f}_{3}:\left(-1, \mathrm{e}^{\pi / 2}-2\right) \rightarrow \mathrm{R}$ and $\mathrm{f}_{4}: \mathrm{R} \rightarrow \mathrm{R}$ be functions defined by

(i) $f_{1}(x)=\sin \left(\sqrt{1-e^{-x^{2}}}\right)$,

(ii) $f_{2}(x)=\left\{\begin{array}{ccc}\frac{|\sin x|}{\tan ^{-1} x} & \text { if } & x \neq 0 \\1 & \text { if } & x=0\end{array}\right.$, where the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,