JEE Advanced 2019 — Paper 2

In a Young's double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1 m . A parallel beam of light of wavelength 600 nm is incident on the slits at angle $\alpha$ as shown in figure. On the screen, the point O is equidistant from the slits and distance PO is 11.0 mm . Which of the following statements(s) is/are correct?

A block of mass 2M is attached to a massless spring with spring-constant k . This block is connected to two other blocks of masses M and 2 M using two massless pulleys and strings. The accelerations of the blocks are $a_{1}, a_{2}$ and $a_{3}$ as shown in the figure. the system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $\mathrm{x}_{0}$. Which of the following option(s) is/are correct? [ $g$ is the acceleration due to gravity. neglect friction]

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle $60^{\circ}$ with vertical ? [ $g$ is the acceleration due to gravity]

A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $L=L_{0}$ the particle speed is $v=v_{0}$. The piston is moved inward at a very low speed V such that $\mathrm{V} \ll \frac{\mathrm{dL}}{\mathrm{L}} \mathrm{V}_{0}$, where dL is the infinitesimal displacement of the piston. which of the following statement(s) is/are correct ?

Three glass cylinders of equal height $\mathrm{H}=30 \mathrm{~cm}$ and same refractive index $\mathrm{n}=1.5$ are placed on a horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same $(R=3 m)$. If $H_{1}, H_{2}$ and $H_{3}$ are the apparent depths of a point X on the bottom of the three cylinders, respectively, the correct statement(s) is/are:

An electric dipole with dipole moment $\frac{p_{0}}{\sqrt{2}}(\hat{i}+\hat{j})$ is held fixed at the origin O in the presence of an uniform electric field of magnitude $\mathrm{E}_{0}$. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are : ( $\epsilon_{0}$ is permittivity of free space. $\mathrm{R} \gg$ dipole size)

A free hydrogen atom after absorbing a photon of wavelength $\lambda_{a}$ gets excited from the state $n=1$ to the state

$n=4$. Immediately after that the electron jumps to $n=m$ state by emitting a photon of wavelength $\lambda_{e}$. Let

the change in momentum of atom due to the absorption and the emission are $\Delta \mathrm{p}_{\mathrm{a}}$ and $\Delta \mathrm{p}_{\mathrm{e}}$. respectively. If

$\frac{\lambda_{a}}{\lambda_{e}}=\frac{1}{5}$, which of the option(s) is /are correct ? [Use hc $=1242 \mathrm{eV} \mathrm{nm} ; 1 \mathrm{~nm}=10^{-9} \mathrm{~m}$, h and c are

Planck's constant and speed of light, respectively]

A mixture of ideal gas contains 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure

$P_{0}$, volume $V_{0}$, and temperature $T_{0}$. If the gas mixture is adiabatically compressed to a volume $\frac{V_{0}}{4}$, then the

correct statement(s) is /are, (Given $2^{1.2}=2.3 ; 2^{3.2}=9.2 ; \mathrm{R}$ is gas constant)

Suppose a ${ }_{88}^{226} \mathrm{Ra}$ nucleus at rest and in ground state undergoes $\alpha$-decay to a ${ }_{86}^{222} \mathrm{Rn}$ nucleus in its excited

state. The kinetic energy of the emitted $\alpha$ particle is found to be $4.44 \mathrm{MeV} .{ }_{86}^{222} \mathrm{Rn}$ nucleus then goes to its

ground state by $\gamma$-decay. The energy of the emitted $\gamma$ photon is $\_\_\_$ keV. [Given: atomic mass of

${ }_{88}^{226} \mathrm{Ra}=226.005 \mathrm{u}$, atomic mass of ${ }_{86}^{222} \mathrm{Rn}=222.000 \mathrm{u}$, atomic mass of $\alpha$ particle

$=4.000 \mathrm{u}, 1 \mathrm{u}=931 \mathrm{MeV} / \mathrm{c}^{2}, \mathrm{c}$ is speed of the light]

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $\mathrm{u}_{0}$. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $\mathrm{V}_{1}$. After hitting the ground, the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_{0} / \alpha$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8 \mathrm{~V}_{1}$, the value of $\alpha$ is $\_\_\_\_$ —.

A 10 cm long perfectly conducting wire PQ is moving with a velocity $1 \mathrm{~cm} / \mathrm{s}$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $\mathrm{L}=1 \mathrm{mH}$ and a resistance $\mathrm{R}=1 \Omega$ as shown in the figure. The horizontal rails, L and R lie in the same plane with a uniform magnetic field $B=1 \mathrm{~T}$ perpendicular to the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is $\mathrm{x} \times 10^{-3} \mathrm{~A}$, where the value of x is $\_\_\_\_$ -. [Assume the velocity of wire PQ remains constant ( $1 \mathrm{~cm} / \mathrm{s}$ ) after key S is closed. Given: $\mathrm{e}^{-1}=0.37$, where e is base of the natural logarithm]

A perfectly reflecting mirror of mass $M$ mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\frac{4 \pi \mathrm{M} \Omega}{\mathrm{h}}=10^{24} \mathrm{~m}^{-2}$ with h as Planck's constant. N photons of wavelength $\lambda=8 \pi \times 10^{-6} \mathrm{~m}$ strike the mirror simultaneously at normal incidence such that the mirror gets displaced by $1 \mu \mathrm{~m}$. If the value of N is $\mathrm{x} \times 10^{12}$, then the value of x is $\_\_\_\_$ -. [Consider the spring as massless]

A monochromatic light is incident from air on a refracting surface of prism of angle $75^{\circ}$ and refractive index $n_{0}=\sqrt{3}$. The other refracting surface of the prism is coated by a thin film of material of refractive index $n$ as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $\theta \leq 60^{\circ}$. The value of $n^{2}$ is $\_\_\_\_$ —.

An optical bench has 1.5 m long scale having four equal divisions in each cm . While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is $\_\_\_\_$ —.

Consider the following reactions (unbalanced)

Choose the correct option(s)

The ground state energy of hydrogen atom is -13.6 eV . Consider an electronic state $\psi$ of $\mathrm{He}^{+}$whose energy, azumuthal quantum number and magnetic quantum number are $-3.4 \mathrm{eV}, 2$ and 0 , respectively. Which of the following statement(s) is(are) true for the state $\Psi$ ?

With reference to aqua regia, choose the correct option(s)

The mole fraction of urea in an aqueous solution containing $900\ \mathrm{g}$ of water is $0.05$. If the density of the solution is $1.2\ \mathrm{g\ cm^{-3}}$, the molarity of urea solution is $\_\_\_\_\_$.

[Given: molar mass of urea $=60\ \mathrm{g\ mol^{-1}}$, water $=18\ \mathrm{g\ mol^{-1}}$]

The amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. $\mathrm{HNO}_{3}$ to a compound with the highest oxidation state of sulphur is $\_\_\_\_$ $\left(\right.$ Given data : Molar mass of water $\left.=18 \mathrm{~g} \mathrm{~mol}^{-1}\right)$

The decomposition reaction $2 \mathrm{~N}_{2} \mathrm{O}_{5}(\mathrm{~g}) \xrightarrow{\longrightarrow} \rightarrow 2 \mathrm{~N}_{2} \mathrm{O}_{4}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g})$ is started in a closed cylinder

under isothermal isochoric condition at an initial pressure of 1 atm . After $\mathrm{Y} \times 10^{3} \mathrm{~s}$, the pressure inside the

cylinder is found to be 1.45 atm . If the rate constant of the reaction is $5 \times 10^{-4} \mathrm{~s}^{-1}$, assuming ideal gas

behaviour, the value of $\mathbf{Y}$ is $\_\_\_\_$ -

A line $L$ passing through the point $P(1,4,3)$, is perpendicular to both the lines $\dfrac{x-1}{2}=\dfrac{y+3}{1}=-\dfrac{z-2}{4}$ and $\dfrac{x+2}{3}=\dfrac{y-4}{2}=\dfrac{z+1}{-2}$.

If the position vector of point $Q$ on $L$ is $\left(a_{1}, a_{2}, a_{3}\right)$ such that $(P Q)^{2}=357$, then ( $\mathrm{a}+\mathrm{a}_{2}+\mathrm{a}_{3}$ ) can be:

Let $x \in \mathbb{R}$ and let $P = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{bmatrix}, \quad Q = \begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{bmatrix} \quad \text{and } R = PQP^{-1}$ Then which of the following options is/are correct?

For non-negative integers n , let $f(n) = \sum_{k=1}^{n} \frac{ \sin \left( \frac{(k+1)\pi}{n+2} \right) \sin \left( \frac{(k+2)\pi}{n+2} \right)}{\sin^2 \left( \frac{(n+1)\pi}{n+2} \right)}$

Assuming $\cos ^{-1} x$ takes values in $[0, \pi]$, which of the following options is/are correct?

For $a \in \mathbb{R}|a|>1$, let

Then the possible value(s) of a is/are

The function $\dfrac{\sin (x+a)}{\sin (x+b)}$ has no maxima or minima if

where $P_{k}^{T}$ denotes the transpose of the matrix $P_{k}$. Then which of the following options is/are correct?

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that f has

PROPERTY 1 if $\lim _{h \rightarrow 0} \frac{f(\mathrm{~h})-\mathrm{f}(0)}{\sqrt{\mid \mathrm{h\mid}}}$ exists and is finite, and

PROPERTY 2 if $\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h^{2}}$ exists and if finite.

Then, which of the following options is/are correct?

The value of the integral

$\_\_\_\_$

Let $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$ be two vectors．Consider a vector $\vec{c}=\alpha \vec{a}+\beta \vec{b}, \alpha, \beta \in \mathbb{R}$ ．If the projection of $\vec{c}$ on the vector $(\vec{a}+\vec{b})$ is $3 \sqrt{2}$ ，then the minimum value of $(\vec{c}-(\vec{a} \times \vec{b})) \cdot \vec{c}$ equals $\_\_\_\_$

Five persons A，B，C，D and E are seated in a circular arrangement．If each of them is given a hat of one of the three colours red，blue and green，then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is $\_\_\_\_$

Let $|\mathrm{x}|$ denote the number of elements in a set X . Let $\mathrm{S}=\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs (A, B) such that $1 \leq|B|<|A|$, equals $\_\_\_\_$

A musical instrument is made using four different metal strings, $1,2,3$ and 4 with mass per unit length $\mu$, $2 \mu, 3 \mu$ and $4 \mu$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range $L_{0}$ and $2 L_{0}$. It is found that in string- $1(\mu)$ at free length $L_{0}$ and tension $T_{0}$ the fundamental mode frequency is $f_{0}$. List -I gives the above four strings while list -II lists the magnitude of some quantity.

If the tension in each string is $T_{0}$, the correct match for the highest fundamental frequency in $f_{0}$ units will be,

A musical instrument is made using four different metal strings, $1,2,3$ and 4 with mass per unit length $\mu$, $2 \mu, 3 \mu$ and $4 \mu$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range $L_{0}$ and $2 L_{0}$. It is found that in string- $1(\mu)$ at free length $L_{0}$ and tension $T_{0}$ the fundamental mode frequency is $f_{0}$. List -I gives the above four strings while list -II lists the magnitude of some quantity.

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following, List-I contains some quantities for the $\mathrm{n}^{\text {th }}$ orbit of the atom and List-II contains options showing how they depend on $n$.

Which of the following options has the correct combination considering List-I and List-II?

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following, List-I contains some quantities for the $\mathrm{n}^{\text {th }}$ orbit of the atom and List-II contains options showing how they depend on $n$.

Which of the following options has the correct combination considering List-I and List-II?

Let $f(x)=\sin (\pi \cos x)$ and $g(x)=\cos (2 \pi \sin x)$ be two functions defined for $x>0$. Define the

following sets whose elements are written in the increasing order:

List - I contains the set X, Y, Z and W. List - II contains some information regarding these sets.

Which of the following is the only CORRECT combination?

Let $f(x)=\sin (\pi \cos x)$ and $g(x)=\cos (2 \pi \sin x)$ be two functions defined for $x>0$. Define the

following sets whose elements are written in the increasing order:

List - I contains the set X, Y, Z and W. List - II contains some information regarding these sets.

Which of the following is the only CORRECT combination?