JEE Advanced 2018 — Paper 1

The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $V(r)=k r^{2} / 2$, where $k$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $R$ about the point $O$. If $v$ is the speed of the particle and $L$ is the magnitude of its angular momentum about $O$, which of the following statements is (are) true?

Consider a body of mass 1.0 kg at rest at the origin at time $t=0$. A force $\vec{F}=(\alpha t \hat{i}+\beta \hat{j})$ is applied on the body, where $\alpha=1.0 \mathrm{Ns}^{-1}$ and $\beta=1.0 \mathrm{~N}$. The torque acting on the body about the origin at time $t=1.0 \mathrm{~s}$ is $\vec{\tau}$. Which of the following statements is (are) true?

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

In the figure below, the switches $S_{1}$ and $S_{2}$ are closed simultaneously at $t=0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current $I$ in the middle wire reaches its maximum magnitude $I_{\max }$ at time $t=\tau$. Which of the following statements is (are) true?

Two infinitely long straight wires lie in the $x y$-plane along the lines $x= \pm R$. The wire located at $x=+R$ carries a constant current $I_{l}$ and the wire located at $x=-R$ carries a constant current $I_{2}$. A circular loop of radius $R$ is suspended with its centre at $(0,0, \sqrt{3} R)$ and in a plane parallel to the $x y$-plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\hat{j}$ direction. Which of the following statements regarding the magnetic field $\vec{B}$ is (are) true?

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where ${V}$ is the volume and ${T}$ is the temperature). Which of the statements below is (are) true?

Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A}=a \hat{i}$ and $\vec{B}=a(\cos \omega t \hat{i}+\sin \omega t \hat{j})$, where $a$ is a constant and

$\omega=\pi / 6 \mathrm{rad} \mathrm{s} s^{-1}$. If $|\vec{A}+\vec{B}|=\sqrt{3}|\vec{A}-\vec{B}|$ at time $t=\tau$ for the first time, the value of $\tau$, in seconds,

is $\_\_\_\_$ -.

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0 \mathrm{~ms}^{-1}$ and the man behind walks at a speed $2.0 \mathrm{~ms}^{-1}$. A third man is standing at a height 12 m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 Hz . The speed of sound in air is $330 \mathrm{~ms}^{-1}$. At the instant, when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in Hz , heard by the stationary man at this instant, is $\qquad$ -.

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $60^{\circ}$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $(2-\sqrt{3}) / \sqrt{10} s$, then the height of the top of the inclined plane, in metres, is $\_\_\_\_\_$ . Take $g=10 \mathrm{~ms}^{-2}$.

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 N m -1 and the mass of the block is 2.0 kg . Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of $2.0 \mathrm{~m} \mathrm{~s}^{-1}$ collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is $\_\_\_\_\_$ -.

Three identical capacitors $C_{1}, C_{2}$ and $C_{3}$ have a capacitance of $1.0 \mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_{l}$ is then filled completely with a dielectric material of relative permittivity $\epsilon_{r}$. The cell electromotive force (emf) $V_{0}=8 \mathrm{~V}$. First the switch $S_{l}$ is closed while the switch $S_{2}$ is kept open. When the capacitor $C_{3}$ is fully charged, $S_{l}$ is opened and $S_{2}$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_{3}$ is found to be $5 \mu C$. The value of $\epsilon_{r}=$ $\_\_\_\_\_$.

In the $x y$-plane, the region $y>0$ has a uniform magnetic field $B_{1} \hat{k}$ and the region $y<0$ has another uniform magnetic field $B_{2} \hat{k}$. A positively charged particle is projected from the origin along the positive $y$-axis with speed $v_{0}=\pi \mathrm{ms}^{-1}$ at $t=0$, as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the $x$-axis from below for the first time. If $\mathrm{B}_{2}=4 B_{1}$, the average speed of the particle, in $\mathrm{ms}^{-1}$, along the $x$-axis in the time interval $T$ is $\_\_\_\_\_$ .

Sunlight of intensity $1.3 \mathrm{kWm}^{-2}$ is incident normally on a thin convex lens of focal length 20 cm . Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $k W \mathrm{~m}^{-2}$, at a distance 22 cm from the lens on the other side is $\qquad$ .

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_{1}=300 \mathrm{~K}$ and $T_{2}=100 K$, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_{1}$ and $K_{2}$ respectively. If the temperature at the junction of the two cylinders in the steady state is 200 K , then $K_{1} / K_{2}=$ $\_\_\_\_\_$ .

The compound(s) which generate(s) $\mathrm{N}_{2}$ gas upon thermal decomposition below $300^{\circ} \mathrm{C}$ is (are)

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are)

(Atomic numbers: $\mathrm{Fe}=26, \mathrm{Ni}=28$ )

$\mathrm{~B}$ ased on the compounds of group 15 elements, the correct statement(s) is (are)

A reversible cyclic process for an ideal gas is shown below. Here, $P, V$, and $T$ are pressure, volume and temperature, respectively. The thermodynamic parameters $q, w, H$ and $U$ are heat, work, enthalpy and internal energy, respectively. The correct option(s) is (are)

Among the species given below, the total number of diamagnetic species is $\_\_\_\_$ .

H atom, $\mathrm{NO}_{2}$ monomer, $\mathrm{O}_{2}^{-}$(superoxide), dimeric sulphur in vapour phase,

$\mathrm{Mn}_{3} \mathrm{O}_{4},\left(\mathrm{NH}_{4}\right)_{2}\left[\mathrm{FeCl}_{4}\right],\left(\mathrm{NH}_{4}\right)_{2}\left[\mathrm{NiCl}_{4}\right], \mathrm{K}_{2} \mathrm{MnO}_{4}, \mathrm{~K}_{2} \mathrm{CrO}_{4}$

The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by

$\mathrm{NiCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}$ to form a stable coordination compound. Assume that both the reactions are $100 \%$ complete.

If 1584 g of ammonium sulphate and 952 g of $\mathrm{NiCl}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}$ are used in the preparation, the combined weight

(in grams) of gypsum and the nickel-ammonia coordination compound thus produced is $\_\_\_\_$ .

(Atomic weights in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{Cl}=35.5, \mathrm{Ca}=40, \mathrm{Ni}=59$ )

Consider an ionic solid MX with NaCl structure. Construct a new structure ( $\mathbf{Z}$ ) whose unit cell is constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge balance. (i) Remove all the anions ( $\mathbf{X}$ ) except the central one (ii) Replace all the face centered cations (M) by anions ( $\mathbf{( X )}$ (iii) Remove all the corner cations (M) (iv) Replace the central anion (X) with cation (M) The value of $\left(\frac{\text { number of anions }}{\text { number of cations }}\right)$ in $\mathbf{Z}$ is $\qquad$ .

F or the electrochemical cell,

$\operatorname{Mg}(\mathrm{s})\left|\mathrm{Mg}^{2+}(\mathrm{aq}, 1 \mathrm{M}) \| \mathrm{Cu}^{2+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{Cu}(\mathrm{s})$

the standard emf of the cell is 2.70 V at 300 K . When the concentration of $\mathrm{Mg}^{2+}$ is changed to ${x} \mathrm{M}$,

the cell potential changes to 2.67 V at 300 K . The value of ${x}$ is $\_\_\_\_$ .

(given, $\frac{\mathrm{F}}{\mathrm{R}}=11500 \mathrm{~K} \mathrm{~V}^{-1}$, where $F$ is the Faraday constant and $R$ is the gas constant, $\ln (10)=2.30$ )

A closed tank has two compartments $\mathbf{A}$ and $\mathbf{B}$, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old partition is replaced by a new partition which can slide and conduct heat but does NOT allow the gas to leak across (Figure 2), the volume (in $\mathrm{m}^{3}$ ) of the compartment $\mathbf{A}$ after the system attains equilibrium is

Liquids $\mathbf{A}$ and $\mathbf{B}$ form ideal solution over the entire range of composition. At temperature $\mathbf{T}$, equimolar binary solution of liquids $\mathbf{A}$ and $\mathbf{B}$ has vapour pressure 45 Torr. At the same temperature, a new solution of $\mathbf{A}$ and $\mathbf{B}$ having mole fractions $x_{\mathrm{A}}$ and $x_{\mathrm{B}}$, respectively, has vapour pressure of 22.5 Torr. The value of $x_{\mathrm{A}} / x_{\mathrm{B}}$ in the new solution is $\_\_\_\_$ -. (given that the vapour pressure of pure liquid $\mathbf{A}$ is 20 Torr at temperature T )

The solubility of a salt of weak acid ( $\mathbf{A B}$ ) at pH 3 is $\mathbf{Y} \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1}$. The value of $\mathbf{Y}$ is $\_\_\_\_$ .

(Given that the value of solubility product of $\mathbf{A B}\left(\mathrm{K}_{\mathrm{sp}}\right)=2 \times 10^{-10}$ and the value

of ionization constant of HB $\left.\left(K_{a}\right)=1 \times 10^{-8}\right)$

The plot given below shows $\mathrm{P}-\mathrm{T}$ curves (where $P$ is the pressure and $T$ is the temperature) for two solvents $\mathbf{X}$ and $\mathbf{Y}$ and isomolal solutions of NaCl in these solvents. NaCl completely dissociates in both the solvents. On addition of equal number of moles of a non-volatile solute $\mathbf{S}$ in equal amount (in kg ) of these solvents, the elevation of boiling point of solvent $\mathbf{X}$ is three times that of solvent $\mathbf{Y}$. Solute $\mathbf{S}$ is known to undergo dimerization in these solvents. If the degree of dimerization is 0.7 in solvent $\mathbf{Y}$, the degree of dimerization in solvent $\mathbf{X}$ is $\_\_\_\_$ .

For a non-zero complex number $z$, let $\arg (z)$ denote the principal $\operatorname{argument}$ with $-\pi<\arg (z) \leq \pi$. Then, which of the following statement(s) is (are) FALSE ?

In a triangle PQR , let $\angle \mathrm{PQR}=30^{\circ}$ and the sides PQ and QR have lengths $10 \sqrt{3}$ and 10 , respectively. Then, which of the following statement(s) is (are) TRUE ?

Let $P_{1}: 2 x+y-z=3$ and $P_{2}: x+2 y+z=2$ be two planes. Then, which of the following statement(s) is (are) TRUE ?

For the curve $\sin x+\sin y=1$ lying in the first quadrant

$L=\lim _{x \rightarrow 0} x^{\alpha} \frac{d^{2} y}{d x^{2}}$, where L exists and is non zero. Then

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be two non-constant differentiable functions. If

$f^{\prime}(x)=\left(e^{(f(x)-g(x))}\right) g^{\prime}(x)$ for all $x \in R$,

and $f(1)=g(2)=1$, then which of the following statement(s) is (are) TRUE ?

Let $\mathrm{f}:[0, \infty) \rightarrow \mathrm{R}$ be a continuous function such that $\mathrm{f}(\mathrm{x})=1-2 \mathrm{x}+\int_{0}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}-\mathrm{t}} \mathrm{f}(\mathrm{t}) \mathrm{dt}$ for all

$\mathrm{x} \in[0, \infty)$. Then, which of the following statement(s) is (are) TRUE ?

The value of $\left(\left(\log _{2} 9\right)^{2}\right)^{\frac{1}{\log _{2}\left(\log _{2} 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _{4} 7}}$ is $\_\_\_\_$ .

The number of 5 digit numbers which are divisible by 4 , with digits from the set $\{1,2,3,4,5\}$ and the repetition of digits is allowed, is $\qquad$ .

Let X be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11, \ldots$. , and Y be the set consisting of the first 2018 terms of arithmetic progression $9,16,23, \ldots .$. . Then, the number of elements in the set $\mathrm{X} \cup \mathrm{Y}$ is $\_\_\_\_$.

The number of real solutions of the equation

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is $\_\_\_\_$ .

(Here, the inverse trigonometric functions $\sin ^{-1} x$ and $\cos ^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$,

respectively.)

For each positive integer n , let $\mathrm{y}_{\mathrm{n}}=\frac{1}{\mathrm{n}}((\mathrm{n}+1)(\mathrm{n}+2) \ldots(\mathrm{n}+\mathrm{n}))^{1 / \mathrm{n}} .$ For $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. If $\lim _{n \rightarrow \infty} y_{n}=L$, then the value of [L] is $\_\_\_\_$.

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b}=0$. For some $x, y \in R$, let $\vec{c}=x \vec{a}+y \vec{b}+(\vec{a} \times \vec{b})$. If $|\vec{c}|=2$ and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$, then the value of $8 \cos ^{2} \alpha$ is $\_\_\_\_$ .

Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be three non-zero real numbers such that the equation

has two distinct real roots $\alpha$ and $\beta$ with $\alpha+\beta=\frac{\pi}{3}$. Then, the value of $\frac{b}{a}$ is $\_\_\_\_$ .

A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P(0,0), Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $\mathrm{F}_{2}$ takes away the region which lies between the side PQ and a curve of the form $y=x^{n}(n>1)$. If the area of the region taken away by the farmer $F_{2}$ is exactly $30 \%$ of the area of $\triangle P Q R$, then the value of $n$ is $\_\_\_\_\_$ .

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $\left[\epsilon_{0}\right]$ and $\left[\mu_{0}\right]$ stand for dimensions of the permittivity and permeability of free space respectively. [L] and $[T]$ are dimensions of length and time respectively. All the quantities are given in SI units.

The relation between $[E]$ and $[B]$ is

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $\left[\epsilon_{0}\right]$ and $\left[\mu_{0}\right]$ stand for dimensions of the permittivity and permeability of free space respectively. [L] and $[T]$ are dimensions of length and time respectively. All the quantities are given in SI units.

The relation between $[\epsilon_0]$ and $[\mu_0]$ is

Consider the ratio $r = \frac{(1-a)}{(1+a)}$ to be determined by measuring a dimensionless quantity $a$. If the error in the measurement of $a$ is $\Delta a$ ($\Delta a / a \ll 1$), then what is the error $\Delta r$ in determining $r$?

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in

any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first

power of the error. For example, consider the relation $z=x / y$. If the errors in $x, y$ and $z$ are $\Delta x, \Delta y$ and $\Delta z$,

respectively, then

The series expansion for $\left(1 \pm \frac{\Delta y}{y}\right)^{-1}$, to first power in $\Delta y / y$, is $1 \mp(\Delta \mathrm{y} / y)$. The relative errors in

independent variables are always added. So the error in $z$ will be

The above derivation makes the assumption that $\Delta x / x \ll 1, \Delta y / y \ll 1$. Therefore, the higher powers of

these quantities are neglected.

In an experiment the initial number of radioactive nuclei is 3000. It is found that $1000 \pm 40$ nuclei decayed in the first 1.0s. For $|x| \ll 1$, $\ln(1+x) \approx x$ up to first power in $x$. The error $\Delta\lambda$, in the determination of the decay constant $\lambda$, in s$^{-1}$, is

Let E$_1$E$_2$ and F$_1$F$_2$ be the chords of S passing through the point P$_0$(1, 1) and parallel to the x-axis and the y-axis, respectively. Let G$_1$G$_2$ be the chord of S passing through P$_0$ and having slope $-1$. Let the tangents to S at E$_1$ and E$_2$ meet at E$_3$, the tangents to S at F$_1$ and F$_2$ meet at F$_3$, and the tangents to S at G$_1$ and G$_2$ meet at G$_3$. Then, the points E$_3$, F$_3$ and G$_3$ lie on the curve

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then the mid-point of the line segment MN must lie on the curve

The probability that, on the examination day, the student S$_1$ gets the previously allotted seat R$_1$, and NONE of the remaining students gets the seat previously allotted to him/her is

There are five students $S_{1}, S_{2}, S_{3}, S_{4}$ and $S_{5}$ in a music class and for them there are five seats $R_{1}, R_{2}, R_{3}, R_{4}$ and $R_{5}$ arranged in a row, where initially the seat $R_{i}$ is allotted to the student $S_{i}, i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats.

For i=1, 2, 3, 4, let T$_i$ denote the event that the students S$_i$ and S$_{i+1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T$_1 \cap$ T$_2 \cap$ T$_3 \cap$ T$_4$ is