IIT JEE 2010 Physics Question Paper with Answer and Solution

$A$ block of mass $m$ is on an inclined plane of angle $\theta$. The coefficient of friction between the block and the plane is $\mu$ and $\tan \theta > \mu$. The block is held stationary by applying a force $P$ parallel to the plane. The direction of force pointing up the plane is taken to be positive. As $P$ is varied from $P_1 = mg(\sin \theta - \mu \cos \theta)$ to $P_2 = mg(\sin \theta + \mu \cos \theta)$,the frictional force $f$ versus $P$ graph will look like:

$A$ thin uniform annular disc (see figure) of mass $M$ has outer radius $4 R$ and inner radius $3 R$. The work required to take a unit mass from point $P$ on its axis to infinity is

$A$ real gas behaves like an ideal gas if its

$A$ point mass of $1 \, kg$ collides elastically with a stationary point mass of $5 \, kg$. After their collision, the $1 \, kg$ mass reverses its direction and moves with a speed of $2 \, m/s$. Which of the following statement(s) is (are) correct for the system of these two masses?
$(A)$ Total momentum of the system is $3 \, kg \cdot m/s$
$(B)$ Momentum of $5 \, kg$ mass after collision is $4 \, kg \cdot m/s$
$(C)$ Kinetic energy of the centre of mass is $0.75 \, J$
$(D)$ Total kinetic energy of the system is $4 \, J$

One mole of an ideal gas in initial state $A$ undergoes a cyclic process $ABCA$,as shown in the figure. Its pressure at $A$ is $P_0$. Choose the correct option$(s)$ from the following:
$(A)$ Internal energies at $A$ and $B$ are the same
$(B)$ Work done by the gas in process $AB$ is $P_0 V_0 \ln 4$
$(C)$ Pressure at $C$ is $\frac{P_0}{4}$
$(D)$ Temperature at $C$ is $\frac{T_0}{4}$

$A$ student uses a simple pendulum of exactly $1 \ m$ length to determine $g$,the acceleration due to gravity. He uses a stopwatch with a least count of $1 \ s$ for this and records $40 \ s$ for $20$ oscillations. For this observation,which of the following statement$(s)$ is (are) true?
$(A)$ Error $\Delta T$ in measuring $T$,the time period,is $0.05 \ s$
$(B)$ Error $\Delta T$ in measuring $T$,the time period,is $1 \ s$
$(C)$ Percentage error in the determination of $g$ is $5 \%$
$(D)$ Percentage error in the determination of $g$ is $2.5 \%$

When a particle of mass $m$ moves on the $x$-axis in a potential of the form $V(x)=kx^2$,it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt{\frac{m}{k}}$,as can be seen easily using dimensional analysis. However,the motion of a particle can be periodic even when its potential energy increases on both sides of $x=0$ in a way different from $kx^2$ and its total energy is such that the particle does not escape to infinity. Consider a particle of mass $m$ moving on the $x$-axis. Its potential energy is $V(x)=\alpha x^4$ $(\alpha>0)$ for $|x|$ near the origin and becomes a constant equal to $V_0$ for $|x| \geq X_0$ (see figure).
$1.$ If the total energy of the particle is $E$,it will perform periodic motion only if
$(A)$ $E < 0$
$(B)$ $E > 0$
$(C)$ $V_0 > E > 0$
$(D)$ $E > V_0$
$2.$ For periodic motion of small amplitude $A$,the time period $T$ of this particle is proportional to
$(A)$ $A \sqrt{\frac{m}{\alpha}}$
$(B)$ $\frac{1}{A} \sqrt{\frac{m}{\alpha}}$
$(C)$ $A \sqrt{\frac{\alpha}{m}}$
$(D)$ $A \sqrt{\frac{\alpha}{m}}$
$3.$ The acceleration of this particle for $|x|>X_0$ is
$(A)$ proportional to $V_0$
$(B)$ proportional to $\frac{V_0}{mX_0}$
$(C)$ proportional to $\sqrt{\frac{V_0}{mX_0}}$
$(D)$ zero
Give the answer for questions $1, 2$ and $3$.

Gravitational acceleration on the surface of a planet is $\frac{\sqrt{6}}{11} g$,where $g$ is the gravitational acceleration on the surface of the earth. The average mass density of the planet is $\frac{2}{3}$ times that of the earth. If the escape speed on the surface of the earth is taken to be $11 \ km/s$,the escape speed on the surface of the planet in $km/s$ will be:

$A$ piece of ice (heat capacity $= 2100 \ J \ kg^{-1} \ ^{\circ}C^{-1}$ and latent heat $= 3.36 \times 10^5 \ J \ kg^{-1}$) of mass $m$ grams is at $-5^{\circ}C$ at atmospheric pressure. It is given $420 \ J$ of heat so that the ice starts melting. Finally,when the ice-water mixture is in equilibrium,it is found that $1 \ g$ of ice has melted. Assuming there is no other heat exchange in the process,the value of $m$ is:

$A$ stationary source is emitting sound at a fixed frequency $f_0$,which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is $1.2 \%$ of $f_0$. What is the difference in the speeds of the cars (in $km/h$) to the nearest integer? The cars are moving at constant speeds much smaller than the speed of sound,which is $330 \ m/s$.

Two spherical bodies $A$ (radius $6 \,cm$) and $B$ (radius $18 \,cm$) are at temperatures $T_1$ and $T_2$,respectively. The maximum intensity in the emission spectrum of $A$ is at $500 \,nm$ and in that of $B$ is at $1500 \,nm$. Considering them to be black bodies,what will be the ratio of the rate of total energy radiated by $A$ to that of $B$?

When two progressive waves $y_1=4 \sin (2 x-6 t)$ and $y_2=3 \sin \left(2 x-6 t-\frac{\pi}{2}\right)$ are superimposed,the amplitude of the resultant wave is

$A$ $0.1 \,kg$ mass is suspended from a wire of negligible mass. The length of the wire is $1 \,m$ and its cross-sectional area is $4.9 \times 10^{-7} \,m^2$. If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency $140 \,rad \,s^{-1}$. If the Young's modulus of the material of the wire is $n \times 10^9 \,N \,m^{-2}$, the value of $n$ is

$A$ binary star system consists of two stars $A$ (mass $M_A = 2.2 M_S$) and $B$ (mass $M_B = 11 M_S$),where $M_S$ is the mass of the Sun. They are separated by a distance $d$ and rotate about their common centre of mass,which is stationary. What is the ratio of the angular momentum of star $A$ to the angular momentum of star $B$ about the centre of mass?

One mole of an ideal gas is taken from $a$ to $b$ along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is $w_s$ and that along the dotted line path is $w_d$,then the integer closest to the ratio $w_d / w_s$ is

$A$ block of mass $2 \ kg$ is free to move along the $x$-axis. It is at rest and from $t=0$ onwards it is subjected to a time-dependent force $F(t)$ in the $x$ direction. The force $F(t)$ varies with $t$ as shown in the figure. The kinetic energy of the block after $4.5 \ s$ is (in $J$)

$A$ Vernier calipers has $1 \,mm$ marks on the main scale. It has $20$ equal divisions on the Vernier scale which match with $16$ main scale divisions. For this Vernier calipers, the least count is (in $\,mm$)

$A$ diatomic ideal gas is compressed adiabatically to $\frac{1}{32}$ of its initial volume. If the initial temperature of the gas is $T_1$ (in Kelvin) and the final temperature is $a T_1$,the value of $a$ is

When liquid medicine of density $\rho$ is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop. We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension $T$ when the radius of the drop is $R$. When the force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
$1.$ If the radius of the opening of the dropper is $r$, the vertical force due to the surface tension on the drop of radius $R$ (assuming $r \ll R$) is
$(A)$ $2 \pi r T$ $(B)$ $2 \pi R T$ $(C)$ $\frac{2 \pi r^2 T}{R}$ $(D)$ $\frac{2 \pi R^2 T}{r}$
$2.$ If $r=5 \times 10^{-4} \, m, \rho=10^3 \, kg \, m^{-3}, g=10 \, m/s^2, T=0.11 \, Nm^{-1}$, the radius of the drop when it detaches from the dropper is approximately
$(A)$ $1.4 \times 10^{-3} \, m$ $(B)$ $3.3 \times 10^{-3} \, m$
$(C)$ $2.0 \times 10^{-3} \, m$ $(D)$ $4.1 \times 10^{-3} \, m$
$3.$ After the drop detaches, its surface energy is
$(A)$ $1.4 \times 10^{-6} \, J$ $(B)$ $2.7 \times 10^{-6} \, J$
$(C)$ $5.4 \times 10^{-6} \, J$ $(D)$ $8.1 \times 10^{-6} \, J$
Give the answer for questions $1, 2$ and $3.$

The key feature of Bohr's theory of the spectrum of the hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find the quantized rotational energy of a diatomic molecule,assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
$1.$ $A$ diatomic molecule has a moment of inertia $I$. By Bohr's quantization condition,its rotational energy in the $n^{\text{th}}$ level $(n=1, 2, 3, \dots)$ is:
$(A) \frac{1}{n^2}\left(\frac{h^2}{8 \pi^2 I}\right)$ $(B) \frac{1}{n}\left(\frac{h^2}{8 \pi^2 I}\right)$ $(C) n\left(\frac{h^2}{8 \pi^2 I}\right)$ $(D) n^2\left(\frac{h^2}{8 \pi^2 I}\right)$
$2.$ It is found that the excitation frequency from the ground state $(n=1)$ to the first excited state $(n=2)$ of rotation for the $CO$ molecule is close to $\frac{4}{\pi} \times 10^{11} \text{ Hz}$. Then the moment of inertia of the $CO$ molecule about its centre of mass is close to (Take $h=2 \pi \times 10^{-34} \text{ Js}$):
$(A) 2.76 \times 10^{-46} \text{ kg m}^2$ $(B) 1.87 \times 10^{-46} \text{ kg m}^2$ $(C) 4.67 \times 10^{-47} \text{ kg m}^2$ $(D) 1.17 \times 10^{-47} \text{ kg m}^2$
$3.$ In a $CO$ molecule,the distance between $C$ (mass $= 12 \text{ a.m.u.}$) and $O$ (mass $= 16 \text{ a.m.u.}$),where $1 \text{ a.m.u.} = \frac{5}{3} \times 10^{-27} \text{ kg}$,is close to:
$(A) 2.4 \times 10^{-10} \text{ m}$ $(B) 1.9 \times 10^{-10} \text{ m}$ $(C) 1.3 \times 10^{-10} \text{ m}$ $(D) 4.4 \times 10^{-11} \text{ m}$
Give the answer for questions $1, 2,$ and $3$.

$A$ hollow pipe of length $0.8 \ m$ is closed at one end. At its open end,a $0.8 \ m$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the string is $50 \ N$ and the speed of sound in air is $320 \ m/s$,the mass of the string is: (in $g$)

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature,$100 \ W$,$60 \ W$ and $40 \ W$ bulbs have filament resistances $R_{100}$,$R_{60}$ and $R_{40}$,respectively,the relation between these resistances is

To verify Ohm's law,a student is provided with a test resistor $R_T$,a high resistance $R_1$,a small resistance $R_2$,two identical galvanometers $G_1$ and $G_2$,and a variable voltage source $V$. The correct circuit to carry out the experiment is

An $AC$ voltage source of variable angular frequency $\omega$ and fixed amplitude $V_0$ is connected in series with a capacitance $C$ and an electric bulb of resistance $R$ (inductance zero). When $\omega$ is increased,

$A$ thin flexible wire of length $L$ is connected to two adjacent fixed points and carries a current $I$ in the clockwise direction,as shown in the figure. When the system is put in a uniform magnetic field of strength $B$ going into the plane of the paper,the wire takes the shape of a circle. The tension in the wire is

Consider a thin square sheet of side $L$ and thickness $t$,made of a material of resistivity $\rho$. The resistance between two opposite faces,shown by the shaded areas in the figure is

$A$ few electric field lines for a system of two charges $Q_1$ and $Q_2$ fixed at two different points on the $x$-axis are shown in the figure. These lines suggest that:
$(A)$ $|Q_1| > |Q_2|$
$(B)$ $|Q_1| < |Q_2|$
$(C)$ at a finite distance to the left of $Q_1$ the electric field is zero
$(D)$ at a finite distance to the right of $Q_2$ the electric field is zero

$A$ ray $OP$ of monochromatic light is incident on the face $AB$ of prism $ABCD$ near vertex $B$ at an incident angle of $60^{\circ}$ (see figure). If the refractive index of the material of the prism is $\sqrt{3}$,which of the following is (are) correct?
$(A)$ The ray gets totally internally reflected at face $CD$
$(B)$ The ray comes out through face $AD$
$(C)$ The angle between the incident ray and the emergent ray is $90^{\circ}$
$(D)$ The angle between the incident ray and the emergent ray is $120^{\circ}$

Electrical resistance of certain materials,known as superconductors,changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature $T_c(0)$. An interesting property of superconductors is that their critical temperature becomes smaller than $T_c(0)$ if they are placed in a magnetic field,i.e.,the critical temperature $T_c(B)$ is a function of the magnetic field strength $B$. The dependence of $T_c(B)$ on $B$ is shown in the figure.
$1.$ In the graphs below,the resistance $R$ of a superconductor is shown as a function of its temperature $T$ for two different magnetic fields $B_1$ (solid line) and $B_2$ (dashed line). If $B_2 > B_1$,which of the following graphs shows the correct variation of $R$ with $T$ in these fields?
$2.$ $A$ superconductor has $T_c(0) = 100 \ K$. When a magnetic field of $7.5 \ T$ is applied,its $T_c$ decreases to $75 \ K$. For this material,one can definitely say that when:
$(A)$ $B = 5 \ T, T_c(B) = 80 \ K$
$(B)$ $B = 5 \ T, 75 \ K < T_c(B) < 100 \ K$
$(C)$ $B = 10 \ T, 75 \ K < T_c < 100 \ K$
$(D)$ $B = 10 \ T, T_c = 70 \ K$
Give the answer for questions $1$ and $2$.

The focal length of a thin biconvex lens is $20 \,cm$. When an object is moved from a distance of $25 \,cm$ in front of it to $50 \,cm$, the magnification of its image changes from $m_{25}$ to $m_{50}$. The ratio $\frac{m_{25}}{m_{50}}$ is

An $\alpha$-particle and a proton are accelerated from rest by a potential difference of $100 \ V$. After this,their de Broglie wavelengths are $\lambda_\alpha$ and $\lambda_{p}$ respectively. The ratio $\frac{\lambda_{p}}{\lambda_\alpha}$,to the nearest integer,is

When two identical batteries of internal resistance $1 \Omega$ each are connected in series across a resistor $R$,the rate of heat produced in $R$ is $J_1$. When the same batteries are connected in parallel across $R$,the rate is $J_2$. If $J_1 = 2.25 J_2$,then the value of $R$ in $\Omega$ is

$A$ uniformly charged thin spherical shell of radius $R$ carries a uniform surface charge density of $\sigma$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to

$A$ tiny spherical oil drop carrying a net charge $q$ is balanced in still air with a vertical uniform electric field of strength $\frac{81 \pi}{7} \times 10^5 \text{ Vm}^{-1}$. When the field is switched off,the drop is observed to fall with a terminal velocity $2 \times 10^{-3} \text{ ms}^{-1}$. Given $g = 9.8 \text{ ms}^{-2}$,viscosity of the air $\eta = 1.8 \times 10^{-5} \text{ Ns m}^{-2}$,and the density of oil $\rho = 900 \text{ kg m}^{-3}$,the magnitude of $q$ is:

$A$ biconvex lens of focal length $15 \,cm$ is in front of a plane mirror. The distance between the lens and the mirror is $10 \,cm$. $A$ small object is kept at a distance of $30 \,cm$ from the lens. The final image is

$A$ large glass slab $(\mu = 5/3)$ of thickness $8 \text{ cm}$ is placed over a point source of light on a plane surface. It is observed that light emerges from the top surface of the slab through a circular area of radius $R \text{ cm}$. What is the value of $R$?

An object approaches a convex mirror of radius of curvature $20 \,m$ along its optical axis. The image is observed to move from $\frac{25}{3} \,m$ to $\frac{50}{7} \,m$ from the mirror in $30 \,s$. What is the speed of the object in $km/h$?

At time $t=0$,a battery of $10 \ V$ is connected across points $A$ and $B$ in the given circuit. If the capacitors have no charge initially,at what time (in seconds) does the voltage across them become $4 \ V$? [Take $\ln 5=1.6, \ln 3=1.1$]

Two transparent media of refractive indices $\mu_1$ and $\mu_3$ have a solid lens-shaped transparent material of refractive index $\mu_2$ between them as shown in the figures in Column $II$. $A$ ray traversing these media is also shown in the figures. In Column $I$,different relationships between $\mu_1, \mu_2$,and $\mu_3$ are given. Match them to the ray diagram shown in Column $II$.

You are given many resistances,capacitors and inductors. These are connected to a variable $DC$ voltage source (the first two circuits) or an $AC$ voltage source of $50 \ Hz$ frequency (the next three circuits) in different ways as shown in Column $II$. When a current $I$ (steady state for $DC$ or rms for $AC$) flows through the circuit,the corresponding voltage $V_1$ and $V_2$ (indicated in circuits) are related as shown in Column $I$. Match the two.