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Two identical containers A and B with frictionless pistons contain the same ideal gas at the same temperature and the same volume V. The mass of the gas in A is $m_A$ and that in B is $m_B$. The gas in each cylinder is now allowed to expand isothermally to the same final volume 2V. The changes in the pressure in A and B are found to be $△P$ and 1.5 $△P$ respectively. Then

A body sliding on a smooth inclined plane requires $4\text{s}$ to reach the bottom, starting from rest at the top. How much time does it take to cover one-fourth of the distance starting from rest at the top?

If the ring performs combined rotational and translational motion on a frictionless horizontal surface as shown in figure, find the magnitude of the velocity at a point $P$ on the ring.

Find the maximum force that should act on the $7\text{kg}$ body for the system to accelerate together with $1{\text{m/s}}^2$. Take $g=10{\text{m/s}}^2$.

A pendulum clock keeps correct time at $0^{\circ }C$. Its mean coefficient of linear expansions is $\alpha {/}^{\circ }C$, then the loss, in seconds, per day by the clock if the temperature rises by $t^{\circ }C$ is -

A projectile is fired at an angle of ${45}^o$ with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection is

A ball strikes a wall with a velocity $\overrightarrow{u}$ at an angle $\theta$ with the normal to the wall surface and rebounds from it at an angle $\beta$ with the surface. Then:

Two gases of equal mass are in thermal equilibrium. If $P_a,P_b$ and $V_a$ and $V_b$ are their respective pressures and volumes, then which relation is true

A particle is executing linear SHM of amplitude $A$ and time period $T$. If $v$ refers to the average speed of the particle during any time interval of $\frac{T}{3}$, then the maximum possible value of $v$ in terms of $A$ is

Two metal wires of identical dimensions are connected in series. If ${\sigma }_1$ and ${\sigma }_2$ are the conductivities of the metals respectively, the effective conductivity of the combination is

In a Carnot heat engine, the temperature of source and sink are $500\text{K}$ and $375\text{K}$. If the engine consumes $25\times {10}^5\text{J}$ per cycle, then work done by engine per cycle is

What is the centripetal force experienced by a bike of weight $800\text{N}$ moving at a velocity $v\text{m/s}$ and bending with an angle of ${60}^{\circ }$ with the ground?

Two concentric rings, one with radius of $0.5\text{m}$ and mass $1\text{kg}$ and other having a radius of $1\text{m}$ and mass $2\text{kg}$ are placed together on a horizontal table. Find out net moment of inertia of the system, about an axis perpendicular to their plane and passing through their common centre.

Find the reading of the spring balance if it is assumed to be of negligible mass.

(Take $g=10m/s^2$)

Photons of energy 6 eV are incident on a metal surface whose work function is 4 eV. The minimum kinetic energy of the emitted photo-electrons will be

For a particle moving on a circular path of radius $5cm$, the velocity at an instant ‘$t$’ is $3\hat{i}+4\hat{j}m/s$. Assuming the particle to be in uniform circular motion, the value of centripetal acceleration is

A sphere of radius R has a uniform distribution of electric charge in its volume. At a distance x from its centre, for x < R , the electric field is directly proportional to

The frequency of the incident light falling on a photosensitive metal plate is doubled, the kinetic energy of the emitted photoelectrons is

A rocket is launched with gas ejection speed of $400\text{m/s}$ and acceleration of $25{\text{m/s}}^2$. If mass of the rocket is $4000\text{kg}$, what will be the rate of consumption of the fuel?

(Take $g=10{\text{m/s}}^2$)

A boy walks to his school at a distance of 6 km with constant speed of2.5 $\frac{km}{hr}$ and walks back with a constant speed of 4 $\frac{km}{hr}$. His average speed for round trip expressed in $\frac{km}{hr}$, is

A body moves due East with velocity 20 km/hour and then due North with velocity 15 km/hour. The resultant velocity

A ball is thrown up from the top of a tower at 20 m/s velocity. It took the ball 6 sec to fall to the bottom. How high is the tower?

Two capacitors each of 1μF capacitance are connected in parallel and are then charged by 200V d.c. supply. The total energy of their charges (in joules) is

Two strings $A$ and $B$, made of same material, are stretched by same tension. The radius of string $A$ is double the radius of $B$. A transverse wave travels on $A$ with speed $v_A$ and on $B$ with speed $v_B$. The ratio $\left(\frac{v_A}{v_B}\right)$ is

Two uniform rods of equal length but of different masses are rigidly combined to form a $L-$ shaped body, which is then pivoted about $O$ as shown in figure. If combination is in equilibrium then ratio $M/m$ will be

A pulley-block system as shown in the figure is released from rest. What is the magnitude of acceleration of $\text{COM}$ of pulley-block system? (Take $g=10{\text{m/s}}^2$)

An elevator car whose floor-to-ceiling distance is equal to 2.7 m starts ascending with constant acceleration 1.2 $m/s^2$. 2s after the start, a bolt begins falling from the ceiling of the car. Find:

(a) the bolt's free fall time;

(b) the displacement and the distance covered by the bolt during the free fall in the reference frame fixed to the ground.

Two point charges -q and +q are located at points (0,0, -a) and (0,0,a) respectivley. What is the electric potential at point (0,0, z)? (z > a)

Three particles $A,B$ and $C$ of masses $2\text{kg}$, $3\text{kg}$ and $5\text{kg}$ are placed at the vertices of a scalene triangle as shown in the figure. Find the location of $COM$ of the three particle system

A block B moves with a velocity $u$ relative to the wedge $A$. If the velocity of the wedge is $v$ as shown in figure, what should be the value of $\theta$ so that the block $B$ moves vertically as seen from the ground?

If a load of 9 kg is suspended on a wire, the increase in length is \(4.5 mm\). The force constant of the wire is

In figure, the coefficient of static friction between the floor and block $B$ is $0.30$. The coefficient of static friction between the block $B$ and $A$ is $0.15$. The mass of block $A$ is $\frac{m}{2}\text{kg}$ and $B$ is $m\text{kg}$. What is the maximum horizontal force $F$ in $\text{Newtons}$ that can be applied to block $B$, so that the two blocks move together?

A block of mass $m$ is pushed towards a movable wedge of mass $nm$ and height $h$, with a velocity $u$. All surfaces are smooth. The minimum value of $u$ for which the block reaches the top of the wedge is -

Starting form rest a particle is first accelerated for time $t_1$ with constant acceleration $a_1$ and then stops in time $t_2$ with constant retardation $a_2$. Let $v_1$ be the average velocity in this case and $s_1$ be the total displacement. In the second case, it is accelerated for the same time $t_1$ with constant acceleration $2a_1$ and comes to rest with constant retardation $a_2$ in time $t_3$. If $v_2$ is the average velocity in this case and $s_2$ the total displacement. Then

The ratio of time taken by a body to slide down a smooth surface to rough identical inclined surface of angle $\theta$ is $‘n^{\prime }$. Find the coefficient of friction.

An object $A$ is moving with $10m/s$ and $B$ is moving with $5m/s$ in the same direction as shown. Initially $A$ is $100m$ behind $B$ as shown. Find the time taken by $A$ to meet $B$

Two friends start a race, and together they run for $50\text{m}$. Rohan turns right and runs $60\text{m}$ while Rocky turns left and runs $40\text{m}$. Then Rohan turns left and runs $50\text{m}$ while Rocky turns right and runs $50\text{m}$. How far are the two friends now from each other?

A force acts on a 30 gm particle in such a way that the position of the particle as a function of time is given by $x=3t-4t^2+t^3$, where x is in metres and t is in seconds. The work done during the first 4 seconds is

A slender uniform rod of mass M and length L is pivoted at one end so that it can rotate in a vertical plane (see the figure).



There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical, is

A helicopter of mass M is rising vertically upwards with a uniform acceleration a. If the mass of the persons in the helicopter is m, what is the magnitude and direction of the force exerted by the persons on the floor of the helicopter?

Two forces $F_1=(2\hat{i}–5\hat{j}-6\hat{k})\text{N}$ and $F_2=(-\hat{i}+2\hat{j}-\hat{k})\text{N}$ are acting on a body at the points $(1,1,0)$ and $(0,1,2)$ respectively. Find resultant torque acting on the body about point $(-1,0,1)$ in $\left(\text{N-m}\right)$.

In the below figure, a square loop consisting of an inductor of inductance L and resistor of resistance R is placed between two long parallel wires. The two long straight wires have time-varying current of magnitude $I=I_{0}cos\omega t$ $A$, but the directions of current in them are opposite.Magnitude of emf in this circuit only due to flux change associated with two long straight current carrying wires will be

A laser beam can be focussed on an area equal to the square of its wavelength. A He-Ne laser radiates energy at the rate of 1mW and its wavelength is 632.8 nm. The intensity of focussed beam will be

A thin rod of length $L$ is lying along the $x-$ axis with its ends at $x=0$ and $x=L$. Its linear density (mass/length) varies with $x$ as $\lambda =k{\left(\frac{x}{L}\right)}^n$, when $n$ can be zero or any positive number. If the position $x_{\text{cm}}$ of the centre of mass of the rod is plotted against ${}^{\prime }{n}^{\prime }$, which of the following graphs best approximates the dependence of $x_{\text{cm}}$ on $n$?

How does a ammeter differ from an galvanometer?

An iron rod and another of brass, both at ${27}^{\circ }C$ differ in length by ${10}^{-3}m$. The coefficient of linear expansion for iron is $1.1\times {10}^{-5}$${/}^{\circ }C$ and for brass is $1.9\times 10-5$${/}^{\circ }C$. The temperature at which both these rods will have the same length is

A train starts from rest and its speed increases at a constant rate $\alpha$ to $v$ and then remains constant for an interval and finally decreases to zero at a constant rate $\beta$. The total distance travelled by train is $l$. The time taken to complete the journey is $t$. Then,

In a double slit experiment, green light of $5303\stackrel{\circ }{A}$ falls on a double slit having a separation of $19.44\mu m$ and a width of $4.05\mu m$. The number of bright fringes between the first and the second diffraction minima is :