IIT JEE 2015 Physics Question Paper with Answer and Solution

$A$ bullet is fired vertically upwards with velocity $v$ from the surface of a spherical planet. When it reaches its maximum height,its acceleration due to the planet's gravity is $1/4$ of its value at the surface of the planet. If the escape velocity from the planet is $v_{esc} = v \sqrt{N}$,then the value of $N$ is (ignore energy loss due to atmosphere).

Two identical uniform discs roll without slipping on two different surfaces $AB$ and $CD$ (see figure) starting at $A$ and $C$ with linear speeds $v_1$ and $v_2$,respectively,and always remain in contact with the surfaces. If they reach $B$ and $D$ with the same linear speed and $v_1 = 3 \ m/s$,then $v_2$ in $m/s$ is $(g = 10 \ m/s^2)$.

Two spherical stars $A$ and $B$ emit blackbody radiation. The radius of $A$ is $400$ times that of $B$ and $A$ emits $10^4$ times the power emitted from $B$. The ratio $(\lambda_A / \lambda_B)$ of their wavelengths $\lambda_A$ and $\lambda_B$ at which the peaks occur in their respective radiation curves is

Consider a Vernier callipers in which each $1 \ cm$ on the main scale is divided into $8$ equal divisions and a screw gauge with $100$ divisions on its circular scale. In the Vernier callipers,$5$ divisions of the Vernier scale coincide with $4$ divisions on the main scale and in the screw gauge,one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:
$(A)$ If the pitch of the screw gauge is twice the least count of the Vernier callipers,the least count of the screw gauge is $0.01 \ mm$.
$(B)$ If the pitch of the screw gauge is twice the least count of the Vernier callipers,the least count of the screw gauge is $0.005 \ mm$.
$(C)$ If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers,the least count of the screw gauge is $0.01 \ mm$.
$(D)$ If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers,the least count of the screw gauge is $0.005 \ mm$.

Planck's constant $h$,speed of light $c$,and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option$(s)$ is(are):
$(A)$ $M \propto \sqrt{c}$
$(B)$ $M \propto \sqrt{G}$
$(C)$ $L \propto \sqrt{h}$
$(D)$ $L \propto \sqrt{G}$

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $\omega_1$ and $\omega_2$ and have total energies $E_1$ and $E_2$,respectively. The variations of their momenta $p$ with positions $x$ are shown in the figures. If $\frac{a}{b}= n^2$ and $\frac{a}{R}= n$,then the correct equation$(s)$ is(are):
$(A) E_1 \omega_1 = E_2 \omega_2$
$(B) \frac{\omega_2}{\omega_1} = n^2$
$(C) \omega_1 \omega_2 = n^2$
$(D) \frac{E_1}{\omega_1} = \frac{E_2}{\omega_2}$

$A$ ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O$. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant,the angular speed of the system is $\frac{8}{9} \omega$ and one of the masses is at a distance of $\frac{3}{5} R$ from $O$. At this instant,the distance of the other mass from $O$ is

$A$ container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $T$. Assuming the gases are ideal,the correct statement$(s)$ is(are):
$(A)$ The average energy per mole of the gas mixture is $2RT$.
$(B)$ The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt{6/5}$.
$(C)$ The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1/2$.
$(D)$ The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1/\sqrt{2}$.

$A$ particle of unit mass is moving along the $x$-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column $I$ ($a$ and $U_0$ are constants). Match the potential energies in column $I$ to the corresponding statement$(s)$ in column $II$.

Column $I$	Column $II$
$(A) U_1(x) = \frac{U_0}{2} \left[1 - \left(\frac{x}{a}\right)^2\right]^2$	$(P)$ The force acting on the particle is zero at $x = a$.
$(B) U_2(x) = \frac{U_0}{2} \left(\frac{x}{a}\right)^2$	$(Q)$ The force acting on the particle is zero at $x = 0$.
$(C) U_3(x) = \frac{U_0}{2} \left(\frac{x}{a}\right)^2 \exp \left[-\left(\frac{x}{a}\right)^2\right]$	$(R)$ The force acting on the particle is zero at $x = -a$.
$(D) U_4(x) = \frac{U_0}{2} \left[\frac{x}{a} - \frac{1}{3}\left(\frac{x}{a}\right)^3\right]$	$(S)$ The particle experiences an attractive force towards $x = 0$ in the region $|x| < a$.
	$(T)$ The particle with total energy $\frac{U_0}{4}$ can oscillate about the point $x = -a$.

$A$ large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ (see figure). The point masses are connected by a rigid massless rod of length $\ell$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $M$ is at a distance $r = 3\ell$ from $M$,the tension in the rod is zero for $m = k\left(\frac{M}{288}\right)$. The value of $k$ is

The energy of a system as a function of time $t$ is given as $E(t)=A^2 \exp(-\alpha t)$,where $\alpha=0.2 \ s^{-1}$. The measurement of $A$ has an error of $1.25 \%$. If the error in the measurement of time is $1.50 \%$,the percentage error in the value of $E(t)$ at $t=5 \ s$ is

The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as $\rho_A(r) = k \left(\frac{r}{R}\right)$ and $\rho_B(r) = k \left(\frac{r}{R}\right)^5$,respectively,where $k$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $I_A$ and $I_B$,respectively. If $\frac{I_B}{I_A} = \frac{n}{10}$,the value of $n$ is

Four harmonic waves of equal frequencies and equal intensities $I_0$ have phase angles $0, \pi / 3, 2 \pi / 3$ and $\pi$. When they are superposed,the intensity of the resulting wave is $nI_0$. The value of $n$ is

Two spheres $P$ and $Q$ of equal radii have densities $\rho_1$ and $\rho_2$,respectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities $\sigma_1$ and $\sigma_2$ and viscosities $\eta_1$ and $\eta_2$,respectively. They float in equilibrium with the sphere $P$ in $L_1$ and sphere $Q$ in $L_2$ and the string being taut (see figure). If sphere $P$ alone in $L_2$ has terminal velocity $\overrightarrow{V}_{P}$ and $Q$ alone in $L_1$ has terminal velocity $\overrightarrow{V}_{Q}$,then
$(A)$ $\frac{|\overrightarrow{V}_{P}|}{|\overrightarrow{V}_{Q}|}=\frac{\eta_1}{\eta_2}$
$(B)$ $\frac{|\overrightarrow{V}_{P}|}{|\overrightarrow{V}_{Q}|}=\frac{\eta_2}{\eta_1}$
$(C)$ $\overrightarrow{V}_{P} \cdot \overrightarrow{V}_{Q} > 0$
$(D)$ $\overrightarrow{V}_{P} \cdot \overrightarrow{V}_{Q} < 0$

In terms of potential difference $V$,electric current $I$,permittivity $\varepsilon_0$,permeability $\mu_0$,and speed of light $c$,the dimensionally correct equation$(s)$ is(are):
$(A)$ $\mu_0 I^2 = \varepsilon_0 V^2$
$(B)$ $\varepsilon_0 I = \mu_0 V$
$(C)$ $I = \varepsilon_0 cV$
$(D)$ $\mu_0 cI = V$

In plotting stress versus strain curves for two materials $P$ and $Q$,a student by mistake puts strain on the $y$-axis and stress on the $x$-axis as shown in the figure. Then the correct statement$(s)$ is(are):
$(A)$ $P$ has more tensile strength than $Q$
$(B)$ $P$ is more ductile than $Q$
$(C)$ $P$ is more brittle than $Q$
$(D)$ The Young's modulus of $P$ is more than that of $Q$

$A$ spherical body of radius $R$ consists of a fluid of constant density $\rho$ and is in equilibrium under its own gravity. If $P(r)$ is the pressure at a distance $r$ from the center $(r < R)$,then the correct option$(s)$ is(are):
$(A) P(r=0) = P_c$ (maximum pressure at center)
$(B) \frac{P(r=3R/4)}{P(r=2R/3)} = \frac{63}{80}$
$(C) \frac{P(r=3R/5)}{P(r=2R/5)} = \frac{16}{21}$
$(D) \frac{P(r=R/2)}{P(r=R/3)} = \frac{20}{27}$

An ideal monoatomic gas is confined in a horizontal cylinder by a spring-loaded piston (as shown in the figure). Initially,the gas is at temperature $T_1$,pressure $P_1$,and volume $V_1$,and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$,pressure $P_2$,and volume $V_2$. During this process,the piston moves out by a distance $x$. Ignoring the friction between the piston and the cylinder,the correct statement$(s)$ is(are):
$(A)$ If $V_2=2V_1$ and $T_2=3T_1$,then the energy stored in the spring is $\frac{1}{4}P_1V_1$
$(B)$ If $V_2=2V_1$ and $T_2=3T_1$,then the change in internal energy is $3P_1V_1$
$(C)$ If $V_2=3V_1$ and $T_2=4T_1$,then the work done by the gas is $\frac{7}{3}P_1V_1$
$(D)$ If $V_2=3V_1$ and $T_2=4T_1$,then the heat supplied to the gas is $\frac{41}{6}P_1V_1$

Consider a concave mirror and a convex lens (refractive index $=1.5$) of focal length $10 \ cm$ each,separated by a distance of $50 \ cm$ in air (refractive index $=1$) as shown in the figure. An object is placed at a distance of $15 \ cm$ from the mirror. Its erect image formed by this combination has magnification $M_1$. When the setup is kept in a medium of refractive index $7/6$,the magnification becomes $M_2$. The magnitude $\left|\frac{M_2}{M_1}\right|$ is

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y-z$ plane at $z=\frac{\sqrt{3}}{2} a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $A B C D$ lying in the $x-y$ plane with its center at the origin is $\frac{\lambda L }{ n \varepsilon_0}$ (where $\varepsilon_0$ is the permittivity of free space),then the value of $n$ is

Consider a hydrogen atom with its electron in the $n^{\text{th}}$ orbital. An electromagnetic radiation of wavelength $90 \ nm$ is used to ionize the atom. If the kinetic energy of the ejected electron is $10.4 \ eV$,then the value of $n$ is $(hc = 1242 \ eV \ nm)$.

$A$ nuclear power plant supplying electrical power to a village uses a radioactive material of half-life $T$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is $12.5 \%$ of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $n T$ years,then the value of $n$ is

$A$ Young's double slit interference arrangement with slits $S_1$ and $S_2$ is immersed in water (refractive index $\mu_w = 4/3$) as shown in the figure. The positions of maxima on the surface of water are given by $x^2 = p^2 m^2 \lambda^2 - d^2$,where $\lambda$ is the wavelength of light in air (refractive index $\mu_a = 1$),$2d$ is the separation between the slits,and $m$ is an integer. The value of $p$ is

For the photoelectric effect with incident photon wavelength $\lambda$,the stopping potential is $V_0$. Identify the correct variation$(s)$ of $V_0$ with $\lambda$ and $1/\lambda$.

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\lambda$ are kept parallel to each other. In their resulting electric field, point charges $+q$ and $-q$ are kept in equilibrium between them. The point charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are):

Two identical glass rods $S_1$ and $S_2$ (refractive index $= 1.5$) have one convex end of radius of curvature $10 \ cm$. They are placed with the curved surfaces at a distance $d$ as shown in the figure,with their axes (shown by the dashed line) aligned. When a point source of light $P$ is placed inside rod $S_1$ on its axis at a distance of $50 \ cm$ from the curved face,the light rays emanating from it are found to be parallel to the axis inside $S_2$. The distance $d$ is (in $cm$)

$A$ conductor (shown in the figure) carrying constant current $I$ is kept in the $x-y$ plane in a uniform magnetic field $\vec{B}$. If $F$ is the magnitude of the total magnetic force acting on the conductor,then the correct statement$(s)$ is(are):

In an aluminium $(Al)$ bar of square cross-section,a square hole is drilled and is filled with iron $(Fe)$ as shown in the figure. The electrical resistivities of $Al$ and $Fe$ are $2.7 \times 10^{-8} \ \Omega m$ and $1.0 \times 10^{-7} \ \Omega m$,respectively. The electrical resistance between the two faces $P$ and $Q$ of the composite bar is

Match the nuclear processes given in column $I$ with the appropriate option$(s)$ in column $II$.

Column $I$	Column $II$
$A$. Nuclear fusion	$P$. Absorption of thermal neutrons by ${}_{92}^{235}U$
$B$. Fission in a nuclear reactor	$Q$. ${}_{27}^{60}Co$ nucleus
$C$. $\beta$-decay	$R$. Energy production in stars via hydrogen conversion to helium
$D$. $\gamma$-ray emission	$S$. Heavy water
	$T$. Neutrino emission

For a radioactive material,its activity $A$ and rate of change of its activity $R$ are defined as $A = -\frac{dN}{dt}$ and $R = -\frac{dA}{dt}$,where $N(t)$ is the number of nuclei at time $t$. Two radioactive sources $P$ (mean life $\tau$) and $Q$ (mean life $2\tau$) have the same activity at $t = 0$. Their rates of change of activities at $t = 2\tau$ are $R_P$ and $R_Q$,respectively. If $\frac{R_P}{R_Q} = \frac{n}{e}$,then the value of $n$ is

$A$ monochromatic beam of light is incident at $60^{\circ}$ on one face of an equilateral prism of refractive index $n$ and emerges from the opposite face making an angle $\theta(n)$ with the normal (see the figure). For $n=\sqrt{3}$ the value of $\theta$ is $60^{\circ}$ and $\frac{d \theta}{d n}=m$. The value of $m$ is

In the following circuit,the current through the resistor $R(=2 \Omega)$ is $I$ Amperes. The value of $I$ is

$A$ fission reaction is given by ${ }_{92}^{236} U \rightarrow{ }_{54}^{140} Xe +{ }_{38}^{94} Sr + x + y$,where $x$ and $y$ are two particles. Considering ${ }_{92}^{236} U$ to be at rest,the kinetic energies of the products are denoted by $K_{Xe}, K_{Sr}, K_x (2 \ MeV)$ and $K_y (2 \ MeV)$,respectively. Let the binding energies per nucleon of ${ }_{92}^{236} U, { }_{54}^{140} Xe$ and ${ }_{38}^{94} Sr$ be $7.5 \ MeV, 8.5 \ MeV$ and $8.5 \ MeV$ respectively. Considering different conservation laws,the correct option$(s)$ is(are):

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin $O$. In this distribution,a spherical cavity of radius $R_2$,centred at $P$ with distance $OP = a = R_1 - R_2$ (see figure) is made. If the electric field inside the cavity at position $\vec{r}$ is $\vec{E}(\vec{r})$,then the correct statement$(s)$ is(are):

$A$ parallel plate capacitor having plates of area $S$ and plate separation $d$,has capacitance $C_1$ in air. When two dielectrics of different relative permittivities $(\varepsilon_1=2$ and $\varepsilon_2=4)$ are introduced between the two plates as shown in the figure,the capacitance becomes $C_2$. The ratio $\frac{C_2}{C_1}$ is

Light guidance in an optical fiber can be understood by considering a structure comprising of a thin solid glass cylinder of refractive index $n_1$ surrounded by a medium of lower refractive index $n_2$. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media $n_1$ and $n_2$. All rays with the angle of incidence $i$ less than a particular value $i_m$ are confined in the medium of refractive index $n_1$. The numerical aperture $(NA)$ of the structure is defined as $\sin i_m$.
$1.$ For two structures namely $S_1$ with $n_1=\sqrt{45}/4$ and $n_2=3/2$,and $S_2$ with $n_1=8/5$ and $n_2=7/5$,and taking the refractive index of water to be $4/3$ and that of air to be $1$,the correct option$(s)$ is(are):
$(A)$ $NA$ of $S_1$ immersed in water is the same as that of $S_2$ immersed in a liquid of refractive index $\frac{16}{3\sqrt{15}}$
$(B)$ $NA$ of $S_1$ immersed in a liquid of refractive index $\frac{6}{\sqrt{15}}$ is the same as that of $S_2$ immersed in water
$(C)$ $NA$ of $S_1$ placed in air is the same as that of $S_2$ immersed in a liquid of refractive index $\frac{4}{\sqrt{15}}$
$(D)$ $NA$ of $S_1$ placed in air is the same as that of $S_2$ placed in water
$2.$ If two structures of the same cross-sectional area,but different numerical apertures $NA_1$ and $NA_2$ $(NA_2 < NA_1)$ are joined longitudinally,the numerical aperture of the combined structure is:
$(A)$ $\frac{NA_1 NA_2}{NA_1+NA_2}$ $(B)$ $NA_1+NA_2$ $(C)$ $NA_1$ $(D)$ $NA_2$

In a thin rectangular metallic strip, a constant current $I$ flows along the positive $x$-direction, as shown in the figure. The length, width, and thickness of the strip are $\ell$, $w$, and $d$, respectively. A uniform magnetic field $\vec{B}$ is applied on the strip along the positive $y$-direction. Due to this, the charge carriers experience a net deflection along the $z$-direction. This results in the accumulation of charge carriers on the surface $PQRS$ and the appearance of equal and opposite charges on the face opposite to $PQRS$. A potential difference along the $z$-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross-section of the strip and carried by electrons.
$1.$ Consider two different metallic strips ($1$ and $2$) of the same material. Their lengths are the same, widths are $w_1$ and $w_2$, and thicknesses are $d_1$ and $d_2$, respectively. Two points $K$ and $M$ are symmetrically located on the opposite faces parallel to the $x$-$y$ plane (see figure). $V_1$ and $V_2$ are the potential differences between $K$ and $M$ in strips $1$ and $2$, respectively. Then, for a given current $I$ flowing through them in a given magnetic field strength $B$, the correct statement$(s)$ is(are):
$(A)$ If $w_1=w_2$ and $d_1=2d_2$, then $V_2=2V_1$
$(B)$ If $w_1=w_2$ and $d_1=2d_2$, then $V_2=V_1$
$(C)$ If $w_1=2w_2$ and $d_1=d_2$, then $V_2=2V_1$
$(D)$ If $w_1=2w_2$ and $d_1=d_2$, then $V_2=V_1$
$2.$ Consider two different metallic strips ($1$ and $2$) of same dimensions (lengths $\ell$, width $w$, and thickness $d$) with carrier densities $n_1$ and $n_2$, respectively. Strip $1$ is placed in magnetic field $B_1$ and strip $2$ is placed in magnetic field $B_2$, both along positive $y$-directions. Then $V_1$ and $V_2$ are the potential differences developed between $K$ and $M$ in strips $1$ and $2$, respectively. Assuming that the current $I$ is the same for both the strips, the correct option$(s)$ is(are):
$(A)$ If $B_1=B_2$ and $n_1=2n_2$, then $V_2=2V_1$
$(B)$ If $B_1=B_2$ and $n_1=2n_2$, then $V_2=V_1$
$(C)$ If $B_1=2B_2$ and $n_1=n_2$, then $V_2=0.5V_1$
$(D)$ If $B_1=2B_2$ and $n_1=n_2$, then $V_2=V_1$
Give the answer for question $1$ and $2$.

An electron in an excited state of $Li^{2+}$ ion has angular momentum $\frac{3 h}{2 \pi}$. The de-Broglie wavelength of the electron in this state is $p \pi a_{0}$ (where, $a_{0} = \text{Bohr radius}$). The value of $p$ is