JEE Advanced 2019 — Paper 1

Consider a spherical gaseous cloud of mass density $\rho(r)$ in free space where $r$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If $\rho(\mathrm{r})$ is constant in time, the particle number density $\mathrm{n}(\mathrm{r})=\rho(\mathrm{r}) / \mathrm{m}$ is[0pt] [ $G$ is universal gravitational constant]

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is $V_{0}$. A hole with a small area $\alpha 4 \pi R^{2}(\alpha \ll 1)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

A current carrying wire heats a metal rod. The wire provides a constant power ( P ) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $(\mathrm{T})$ in the metal rod changes with time ( t ) as $\mathrm{T}(\mathrm{t})=\mathrm{T}_{0}\left(1+\beta \mathrm{t}^{1 / 4}\right)$ where $\beta$ is a constant with appropriate dimension while $\mathrm{T}_{0}$ is a constant with dimension of temperature. The heat capacity of the metal is

In a radioactive sample ${ }_{19}^{40} \mathrm{~K}$ nuclei either decay into stable ${ }_{20}^{40} \mathrm{Ca}$ nuclei with decay constant $4.5 \times 10^{-10}$ per year or into stable ${ }_{18}^{40} \mathrm{Ar}$ nuclei with decay constant $0.5 \times 10^{-10}$ per year. Given that in this sample all the stable ${ }_{20}^{40} \mathrm{Ca}$ and ${ }_{18}^{40} \mathrm{Ar}$ nuclei are produced by the ${ }_{19}^{40} \mathrm{~K}$ nuclei only. In time $\mathrm{t} \times 10^{9}$ years, if the ratio of the sum of stable ${ }_{20}^{40} \mathrm{Ca}$ and ${ }_{18}^{40} \mathrm{Ar}$ nuclei to the radioactive ${ }_{19}^{40} \mathrm{~K}$ nuclei is 99 , the value of t will be [Given $: \ln 10=2.3$ ]

A conducting wire of parabolic shape, initially $y=x^{2}$, is moving with velocity $\vec{V}=V_{0} \hat{i}$ in a non uniform magnetic field $\overrightarrow{\mathrm{B}}=\mathrm{B}_{0}\left(1+\left(\frac{\mathrm{y}}{\mathrm{L}}\right)^{\beta}\right) \hat{\mathrm{k}}$, as shown in figure. If $\mathrm{V}_{0}$ , $\mathrm{B}_{0}, \mathrm{~L}$ and $\beta$ are positive constants and $\Delta \phi$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

A thin convex lens is made of two materials with refractive indices $n_{1}$ and $n_{2}$, as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. $f$ is the focal length of the lens when $\mathrm{n}_{1}=\mathrm{n}_{2}=\mathrm{n}$. The focal length is $f+$ $\Delta f$ when $\mathrm{n}_{1}=\mathrm{n}$ and $\mathrm{n}_{2}=\mathrm{n}+\Delta \mathrm{n}$. Assuming $\Delta \mathrm{n} \ll(\mathrm{n}-1)$ and $1<\mathrm{n}<2$. The correct statement(s) is/are.

A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ of different materials having water contact angles of $0^{\circ}$ and $60^{\circ}$, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is (are) correct?[0pt] [Surface tension of a water $=0.075 \mathrm{~N} / \mathrm{m}$, density of water $=1000 \mathrm{~kg} / \mathrm{m}^{3}$, take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ ]

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of L , which of the following statement(s) is/are correct?

In the circuit shown, initially there is no charge on capacitors and keys $S_{1}$ and $S_{2}$ are open. The values of the

capacitors are $\mathrm{C}_{1}=10 \mu \mathrm{~F}, \mathrm{C}_{2}=30 \mu \mathrm{~F}$ and $\mathrm{C}_{3}$ $=\mathrm{C}_{4}=80 \mu \mathrm{~F}$. Which of the statement(s) is/are

correct?

One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $(\mathrm{V}-\mathrm{T})$ diagram. The correct statement(s) is/are:

[ R is the gas constant]

Two identical moving coil galvanometers have $10 \Omega$ resistance and full scale deflection at $2 \mu \mathrm{~A}$ current. One of them is converted into a voltmeter of 100 mV full scale reading and the other into an Ammeter of 1 mA full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with $\mathrm{R}=1000 \Omega$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area $0.5 \mathrm{~cm}^{2}$ and, length $\sqrt{3} \mathrm{~m}$ and 1 m , respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are $30^{\circ}$ and $60^{\circ}$, respectively. If elongation in copper wire is $\left(\Delta \ell_{C}\right)$ and elongation in steel wire is $\left(\Delta \ell_{\mathrm{S}}\right)$, then the ratio $\frac{\Delta \ell_{\mathrm{C}}}{\Delta \ell_{\mathrm{S}}}$ is $\_\_\_\_$

[Young's modulus for copper and steel are $1 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$ and $2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$, respectively.]

A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force $\vec{F}=(\alpha y \hat{i}+2 \alpha x \hat{j}) N$, where x and y are in meter and $\alpha=-1 \mathrm{Nm}^{-1}$. The work done on the particle by this force $\vec{F}$ will be $\_\_\_\_$ Joule.

A train S1, moving with a uniform velocity of 108 $\mathrm{km} / \mathrm{h}$, approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of $36 \mathrm{~km} / \mathrm{h}$ towards S 2 , as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz . When O is 600 m away from S2 and distance between S 1 and S 2 is 800 m , the number of beats heard by O is $\_\_\_\_$ . [Speed of the sound $=330 \mathrm{~m} / \mathrm{s}$ ]

A liquid at $30^{\circ} \mathrm{C}$ is poured very slowly into a Calorimeter that is at temperature of $110^{\circ} \mathrm{C}$. The boiling temperature of the liquid is $80^{\circ} \mathrm{C}$. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid the equilibrium temperature is found to be $50^{\circ} \mathrm{C}$. The ratio of the Latent heat of the liquid to its specific heat will be $\_\_\_\_$ ${ }^{\circ} \mathrm{C}$. [Neglect the heat exchange with surrounding]

A planar structure of length L and width W is made of two different optical media of refractive indices $\mathrm{n}_{1}=1.5$ and $\mathrm{n}_{2}=1.44$ as shown in figure. If $\mathrm{L} \gg \mathrm{W}$, a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the structure. For $\mathrm{L}=9.6 \mathrm{~m}$, if the incident angle $\theta$ is varied, the maximum time taken by a ray to exit the plane CD is $\mathrm{t} \times 10^{-9} \mathrm{~s}$, where t is $\_\_\_\_$ . [Speed of light $\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ ]

A parallel plate capacitor of capacitance $C$ has spacing $d$ between two plates having area A. The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickness $\delta=\frac{\mathrm{d}}{\mathrm{N}}$. The dielectric constant of the $m^{\text {th }}$ layer is $K_{m}=K\left(1+\frac{m}{N}\right)$. For a very large $N\left(>10^{3}\right)$, the capacitance $C$ is $\alpha\left(\frac{K \varepsilon_{0} A}{d \ln 2}\right)$. The value of $\alpha$ will be $\_\_\_\_$ . [ $\epsilon_{0}$ is the permittivity of free space]

The green colour produced in the borax bead test of a chromium(III) salt is due to

Calamine, malachite, magnetite and cryolite, respectively, are

Which of the following statements(s) is (are) true ?

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.

Fusion of $\mathrm{MnO}_{2}$ with KOH in presence of $\mathrm{O}_{2}$ produces a salt W . Alkaline solution of W upon electrolytic oxidation yields another salt X . The manganese containing ions present in W and X , respectively, are Y and Z. Correct statement(s) is(are)

Which of the following statement(s) is(are) correct regarding the root mean square speed ( $\mathrm{U}_{\mathrm{rms}}$ ) and average translational kinetic energy ( $\varepsilon_{\mathrm{av}}$ ) of a molecule in a gas at equilibrium?

In the decay sequence,

${ }_{92}^{238} \mathrm{U} \xrightarrow{-x_{1}}{ }_{90}^{234} \mathrm{Th} \xrightarrow{-x_{2}}{ }_{91}^{234} \mathrm{~Pa} \xrightarrow{-x_{3}}{ }^{234} \mathrm{Z} \xrightarrow{-\mathrm{x}_{4}}{ }_{90}^{230} \mathrm{Th}$

$\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}$ and $\mathrm{x}_{4}$ are particles /radiation emitted by the respective isotopes. The correct option(s) is(are)

On dissolving 0.5 g of a non-volatile non-ionic solute to 39 g of benzene, its vapour pressure decreases from 650 mm Hg to 640 mm Hg . The depression of freezing point of benzene (in K ) upon addition of the solute is (Given data: Molar mass and the molal freezing point depression constant of benzene are $78 \mathrm{~g} \mathrm{~mol}^{-1}$ and $5.12 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$, respectively)

$\ldots \ldots \ldots$

The rate of the reaction for $[\mathrm{A}]=0.15 \mathrm{~mol} \mathrm{dm}^{-3},[\mathrm{~B}]=0.25 \mathrm{~mol} \mathrm{dm}^{-3}$ and

$[\mathrm{C}]=0.15 \mathrm{~mol} \mathrm{dm}^{-3}$ is found to be $\mathrm{Y} \times 10^{-5} \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}$. The value of Y is

At 143 K , the reaction of $\mathrm{XeF}_{4}$ with $\mathrm{O}_{2} \mathrm{~F}_{2}$ produces xenon compound Y. The total number of lone Pair(s) of electrons present on the whole molecule of Y is $\_\_\_\_$

Let $M=\left[\begin{array}{cc}\sin ^{4} \theta & -1-\sin ^{2}\\ \theta 1+\cos ^{2} \theta & \cos ^{4} \theta\end{array}\right]=\alpha 1+\beta M^{-1}$, where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are

real numbers, and 1 is the $2 \times 2$ identity matrix. If $\alpha^{*}$ is the minimum of the set $\{\alpha(\theta): \theta \in[0,2 \pi)\}$ and $\beta^{*}$

is the minimum of the set $\{\beta(\theta): \theta \in[0,2 \pi)\}$, then the value of $\alpha^{*}+\beta^{*}$ is

The area of the region $\left\{(x, y): x y \leq 8,1 \leq y \leq x^{2}\right\}$ is

Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i| \geq \sqrt{5}$. If the complex number $z_{0}$ is such that $\frac{1}{\left|z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$, then the principal argument of $\frac{4-z_{0}-\overline{z_{0}}}{z_{0}-\overline{z_{0}}+2 i}$ is

A line $y=m x+1$ intersects the circle $(x-3)^{2}+(y+2)^{2}=25$ at the points $P$ and $Q$. If the midpoint

of the line segment PQ has x -coordinate $-\frac{3}{5}$, then which one of the following options is correct?

Let $\alpha$ and $\beta$ be the roots of $x^{2}-x-1=0$, with $\alpha>\beta$. For all positive integers $n$, define

$a_{n}=\frac{a^{n}-\beta^{n}}{\alpha-\beta}, \quad n \geq 1, b_{1}=1$ and $b_{n}=a_{n-1}+a_{n+1}, n \geq 2$. Then which of the following

options is/are correct?

In a non-right-angled triangle $\Delta \mathrm{PQR}$, let $\mathrm{p}, \mathrm{q}, \mathrm{r}$ denote the lengths of the sides opposite to the angles at P , $\mathrm{Q}, \mathrm{R}$ respectively. The median from R meets the side PQ at S , the perpendicular from P meets the side QR at $E$, and RS and PE intersect at $O$. If $p=\sqrt{3}, q=1$, and the radius of the circumcircle of the $\triangle P Q R$ equals 1 , then which of the following options is/are correct?

Let $\Gamma$ denote a curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_{P}$. If $P Y_{P}$ has length 1 for each point $P$ on $\Gamma$, then which of the following option is/are correct?

Let $\mathrm{M}=\left[\begin{array}{lll}0 & 1 & \mathrm{a}\\ 1 & 2 & 3\\ 3 & \mathrm{~b} & 1\end{array}\right]$ and adj $\mathrm{M}=\left[\begin{array}{ccc}-1 & 1 & -1\\ 8 & -6 & 2\\ -5 & 3 & -1\end{array}\right]$ where $a$ and $b$ area real numbers. Which of the following options is/are correct?

Define the collections $\left\{E_{1}, E_{2}, E_{3}, \ldots \ldots\right\}$ of ellipses and $\left\{R_{1}, R_{2}, R_{3}, \ldots\right\}$ of rectangles as follows:

$E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1$;

$R_{1}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_{1}$;

$E_{n}$ : ellipse $\frac{x^{2}}{a_{n}^{2}}+\frac{y^{2}}{b_{n}^{2}}=1$ of largest area inscribed in $R_{n-1}, n>1$;

$\mathrm{R}_{\mathrm{n}}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $\mathrm{E}_{\mathrm{n}}, \mathrm{n}>1$.

Then which of the following options is/are correct?

Let $f: R \rightarrow R$ by given by $f(x) = \begin{cases} x^5 + 5x^4 + 10x^3 + 10x^2 + 3x + 1, & x < 0; \\ x^2 - x + 1, & 0 \le x < 1; \\ \frac{2}{3}x^3 - 4x^2 + 7x - \frac{8}{3}, & 1 \le x < 3; \\ (x-2)\log_e(x-2) - x + \frac{10}{3}, & x \ge 3. \end{cases}$

Then which of the following options is/are correct?

Let $L_{1}$ and $L_{2}$ denote the lines $\vec{r}=\hat{i}+\lambda(-\hat{i}+2 \hat{j}+2 \hat{k}), \lambda \in R$ and $\vec{r}=\mu(2 \hat{i}-\hat{j}+2 \hat{k}), \mu \in R$

respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, then which of the

following options describe (s) $\mathrm{L}_{3}$ ?

There are three bags $B_{1}, B_{2}$ and $B_{3}$. The bag $B_{1}$ contains 5 red and 5 green balls, $B_{2}$ contains 3 red and 5 green balls and $B_{3}$ contains 5 red and 3 green balls. Bags $B_{1}, B_{2}$ and $B_{3}$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Let $\omega \neq 1$ be a cube root of unity. Then the minimum of the set $\left\{\left|a+b \omega+c \omega^{2}\right|^{2}: a, b, c\right.$ distinct non-zero integers $\}$ equals $\_\_\_\_$

If $I=\frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \frac{d x}{\left(1+e^{\sin x}\right)(2-\cos 2 x)}$ then $27 I^{2}$ equals $\_\_\_\_$

Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{0,1\}$. Let the events $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ be given by $\begin{aligned} & E_{1}=\{A \in S: \operatorname{det} A=0\} \text { and } & E_{2}=\{A \in S: \text { sum of entries of } A \text { is } 7\} \end{aligned}$ If a matrix is chosen at random from $S$, then the conditional probability $P\left(E_{1} / E_{2}\right)$ equals $\_\_\_\_$

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to line $8 x-6 y-23=0$. Let $\Gamma_{A}$ and $\Gamma_{B}$ be circles of radii 2 and 1 with centres A and B respectively. Let $T$ be a common tangent to the circles $\Gamma_{A}$ and $\Gamma_{\mathrm{B}}$ such that both the circles are on the same side of T . If C is the point of intersection of T and the line passing through $A$ and $B$, then the length of the line segment $A C$ is $\_\_\_\_$ -

Let $\mathrm{AP}(\mathrm{a} ; \mathrm{d})$ denote the set of all the terms of an infinite arithmetic progression with first term a and common

difference $\mathrm{d}>0$. If $\mathrm{AP}(1 ; 3) \cap \operatorname{AP}(2 ; 5) \cap \mathrm{AP}(3 ; 7)=\mathrm{AP}(\mathrm{a} ; \mathrm{d})$ then $\mathrm{a}+\mathrm{d}$ equals $\_\_\_\_$

Three lines are given by $\vec{r}=\lambda \hat{i}, \lambda \in R, \vec{r}=\mu(\hat{i}+\hat{j}), \mu \in R$ and $\vec{r}=v(\hat{i}+\hat{j}+\hat{k}), v \in R$. Let the

lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is

$\Delta$ then the value of $(6 \Delta)^{2}$ equals