An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are 6.4 cm and 4.8 cm, respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is:
(1) $\frac { 9 } { 16 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 81 } { 256 }$

A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of $5.00\,\mathrm{cm\,s^{-1}}$. Its kinetic energy is:
(1) $8.75 \times 10^{-4}\,\mathrm{J}$
(2) $8.75 \times 10^{-3}\,\mathrm{J}$
(3) $6.25 \times 10^{-4}\,\mathrm{J}$
(4) $1.13 \times 10^{-3}\,\mathrm{J}$

Two liquids of densities $\rho_{1}$ and $\rho_{2}$ ($\rho_{2} = 2\rho_{1}$) are filled up behind a square wall of side 10 m as shown in figure. Each liquid has a height of 5 m. The ratio of the forces due to these liquids exerted on upper part MN to that at the lower part NO is (Assume that the liquids are not mixing):
(1) $\frac{1}{3}$
(2) $\frac{2}{3}$
(3) $\frac{1}{2}$
(4) $\frac{1}{4}$

A body is moving in a low circular orbit about a planet of mass $M$ and radius $R$. The radius of the orbit can be taken to be $R$ itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is:
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) 2
(3) 1
(4) $\sqrt { 2 }$

Consider a solid sphere of radius $R$ and mass density $\rho ( r ) = \rho _ { 0 } \left( 1 - \frac { r ^ { 2 } } { R ^ { 2 } } \right) , 0 < r \leq R$. The minimum density of a liquid in which it will float is:
(1) $\frac { \rho _ { 0 } } { 3 }$
(2) $\frac { \rho _ { 0 } } { 5 }$
(3) $\frac { 2 \rho _ { 0 } } { 5 }$
(4) $\frac { 2 \rho _ { 0 } } { 3 }$

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$. The temperature of the hot reservoir of the first engine is $\mathrm { T } _ { 1 }$ and the temperature of the cold reservoir of the second engine is $T _ { 2 }$. $T$ is temperature of the sink of first engine which is also the source for the second engine. How is T related to $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$, if both the engines perform equal amount of work?
(1) $\mathrm { T } = \frac { 2 \mathrm {~T} _ { 1 } \mathrm {~T} _ { 2 } } { \mathrm {~T} _ { 1 } + \mathrm { T } _ { 2 } }$
(2) $\mathrm { T } = \frac { \mathrm { T } _ { 1 } + \mathrm { T } _ { 2 } } { 2 }$
(3) $T = \sqrt { T _ { 1 } T _ { 2 } }$
(4) $\mathrm { T } = 0$

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period $T_1$ and (ii) back and forth in a direction perpendicular to its plane, with a period $T_2$. The ratio $\frac{T_1}{T_2}$ will be:
(1) $\frac{2}{\sqrt{3}}$
(2) $\frac{2}{3}$
(3) $\frac{3}{\sqrt{2}}$
(4) $\frac{\sqrt{2}}{3}$

A stationary observer receives sound from two identical tuning forks, one of which approaches and the other one recedes with the same speed (much less than the speed of sound). The observer hears 2 beats/sec. The oscillation frequency of each tuning fork is $v _ { 0 } = 1400 \mathrm {~Hz}$ and the velocity of sound in air is $350 \mathrm {~m} / \mathrm { s }$. The speed of each tuning fork is close to:
(1) $\frac { 1 } { 2 } \mathrm {~m} / \mathrm { s }$
(2) $1 \mathrm {~m} / \mathrm { s }$
(3) $\frac { 1 } { 4 } \mathrm {~m} / \mathrm { s }$
(4) $\frac { 1 } { 8 } \mathrm {~m} / \mathrm { s }$

An emf of 20 V is applied at time $t = 0$ to a circuit containing in series 10 mH inductor and $5 \Omega$ resistor. The ratio of the currents at time $t = \infty$ and at $t = 40 \mathrm {~s}$ is close to: (Take $e ^ { 2 } = 7.389$)
(1) 1.06
(2) 1.15
(3) 1.46
(4) 0.84

A thin lens made of glass (refractive index $= 1.5$) of focal length $f = 16 \mathrm {~cm}$ is immersed in a liquid of refractive index 1.42. If its focal length in liquid is $f _ { l }$, then the ratio $f _ { l } / f$ is closest to the integer:
(1) 1
(2) 9
(3) 5
(4) 17

In a Young's double slit experiment, the separation between the slits is 0.15 mm. In the experiment, a source of light of wavelength 589 nm is used and the interference pattern is observed on a screen kept 1.5 m away. The separation between the successive bright fringes on the screen is:
(1) 6.9 mm
(2) 3.9 mm
(3) 5.9 mm
(4) 4.9 mm

An electron (of mass $m$) and a photon have the same energy $E$ in the range of a few eV. The ratio of the de Broglie wavelength associated with the electron and the wavelength of the photon is ($c =$ speed of light in vacuum)
(1) $\frac { 1 } { c } \left( \frac { 2 E } { m } \right) ^ { 1 / 2 }$
(2) $c ( 2 m E ) ^ { 1 / 2 }$
(3) $\frac { 1 } { c } \left( \frac { E } { 2 m } \right) ^ { 1 / 2 }$
(4) $\left( \frac { E } { 2 m } \right) ^ { 1 / 2 }$

The activity of a radioactive sample falls from $700 \mathrm {~s} ^ { - 1 }$ to $500 \mathrm {~s} ^ { - 1 }$ in 30 minutes. Its half life is close to:
(1) 72 min
(2) 62 min
(3) 66 min
(4) 52 min

A non-isotropic solid metal cube has coefficients of linear expansion as: $5 \times 10 ^ { - 5 } / { } ^ { \circ } \mathrm { C }$ along the x-axis and $5 \times 10 ^ { - 6 } / { } ^ { \circ } \mathrm { C }$ along the y and the z-axis. If the coefficient of volume expansion of the solid is $\mathrm { C } \times 10 ^ { - 6 } / { } ^ { \circ } \mathrm { C }$ then the value of $\mathbf { C }$ is $\_\_\_\_$

An asteroid is moving directly towards the centre of the earth. When at a distance of $10R$ ($R$ is the radius of the earth) from the centre of the earth, it has a speed of $12\,\mathrm{km\,s^{-1}}$. Neglecting the effect of earth's atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is $11.2\,\mathrm{km\,s^{-1}}$)? Give your answer to the nearest integer in $\mathrm{km\,s^{-1}}$ $\underline{\hspace{1cm}}$.

The change in the magnitude of the volume of an ideal gas when a small additional pressure $\Delta \mathrm { P }$ is applied at a constant temperature, is the same as the change when the temperature is reduced by a small quantity $\Delta \mathrm { T }$ at constant pressure. The initial temperature and pressure of the gas were 300 K and 2 atm respectively. If $| \Delta \mathrm { T } | = \mathrm { C } | \Delta \mathrm { P } |$ then value of C in (K/atm) is $\_\_\_\_$

A Carnot engine operates between two reservoirs of temperatures 900 K and 300 K. The engine performs 1200 J of work per cycle. The heat energy (in J) delivered by the engine to the low temperature reservoir, in a cycle, is $\_\_\_\_$

Three containers $C_{1}$, $C_{2}$ and $C_{3}$ have water at different temperatures. The table below shows the final temperature $T$ when different amounts of water (given in liters) are taken from each container and mixed (assume no loss of heat during the process)

$C_{1}$	$C_{2}$	$C_{3}$	$T$
$1l$	$2l$	--	$60^{\circ}\mathrm{C}$
--	$1l$	$2l$	$30^{\circ}\mathrm{C}$
$2l$	--	$1l$	$60^{\circ}\mathrm{C}$
$1l$	$1l$	$1l$	$\theta$

The value of $\theta$ (in ${}^{\circ}\mathrm{C}$ to the nearest integer) is $\underline{\hspace{1cm}}$

Nitrogen gas is at $300^\circ\mathrm{C}$ temperature. The temperature (in K) at which the rms speed of a $\mathrm{H_2}$ molecule would be equal to the rms speed of a nitrogen molecule, is $\_\_\_\_$ (Molar mass of $\mathrm{N_2}$ gas 28 g).

A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is $300 \mathrm {~m} / \mathrm { s }$, the frequency difference between the fundamental and second harmonic of this pipe is $\_\_\_\_$ Hz.

A loop $ABCDEFA$ of straight edges has six corner points $A(0,0,0), B(5,0,0), C(5,5,0), D(0,5,0), E(0,5,5)$ and $F(0,0,5)$. The magnetic field in this region is $\vec { B } = (3 \hat { i } + 4 \hat { k }) T$. The quantity of flux through the loop $ABCDEFA$ (in Wb) is $\_\_\_\_$

A 60 pF capacitor is fully charged by a 20 V supply. It is then disconnected from the supply and is connected to another uncharged 60 pF capacitor in parallel. The electrostatic energy that is lost in this process by the time the charge is redistributed between them is (in nJ) $\_\_\_\_$

The series combination of two batteries, both of the same emf 10 V, but different internal resistance of $20\,\Omega$ and $5\,\Omega$, is connected to the parallel combination of two resistors $30\,\Omega$ and $x\,\Omega$. The voltage difference across the battery of internal resistance $20\,\Omega$ is zero, the value of $x$ (in $\Omega$) is $\underline{\hspace{1cm}}$

The distance between an object and a screen is 100 cm. A lens can produce real image of the object on the screen for two different positions between the screen and the object. The distance between these two positions is 40 cm. If the power of the lens is close to $\left( \frac { \mathrm { N } } { 100 } \right) \mathrm { D }$ where N is an integer, the value of N is $\_\_\_\_$

A prism of angle $A = 1^\circ$, $\mu = 1.5$. A good estimate for the minimum angle of deviation (in degrees) is close to $\frac{N}{10}$. Value of N is $\_\_\_\_$