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The radioactivity of a sample is R₁ at a time T₁ and R₂ at a time T₂. If the half-life of the specimen is T, the number of atoms that have disintegrated in the time (T₂-T₁) is proportional to:

A charged particle of mass ‘m’ and charge ‘q’ is released from rest in an electric field of uniform strength E.
The kinetic energy of the particle after time ‘t’ is ___

Given current of 4 A flows through an inductor of 2 H. What is the energy stored in inductor?

The speed at the maximum height of a projectile is half of its initial speed u. Find the range on the horizontal plane.

The blocks are of mass $2\text{kg}$ shown, is in equilibrium. At $t=0$ right spring in figure (i) and right string in figuree (ii) breaks. Find the ratio of instantaneous acceleration of blocks in fig (i) and fig (ii) (Take $g=10m/s^2$)

The equation of stationary wave along a stretched string is given by

y = 5 sin $\frac{\pi x}{3}cos40\pi t$ , where x and y are in cm and t in second. The

separation between two adjacent nodes is

[CPMT 1990; MP PET 1999; AMU 1999;

DPMT 2004; BHU 2005]

The vapour pressure of water at 20 degree celsius is 17.54 mmHg. What will be the vapour pressure of the water in the apparatus shown after the piston is lowered, decreasing the volume of the gas above the liquid to one half of its initial volume (assume temperature constant).

A) 8.77 mmHg

B)17.54mmHg

C)35.08 mmHg

D) between 8.77 and 17.54mmHg

A particle is moving in a straight line with initial velocity $u$ and uniform acceleration $f$. If the sum of the distances travelled in $t^{th}$ and $(t+1{)}^{th}$ seconds is $100\text{cm}$, then its velocity after $t$ seconds (in $cm/s$) is

An object starts moving from point P to point R via Q as shown in figure. Correct image movement graph is

Two bodies M and N of equal masses are suspended from two separate massless springs of force constants $k_1$ and $k_2$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude M to that of N is

The displacement equation for a particle in standing wave is represented as $y(x,t)=0.4\sin \left(0.5x\right)\cos \left(30t\right)$, where $x$ and $y$ are in $\text{cm}$.
Magnitude of velocity of particle at $x=2.4\text{cm}$ and $t=0.8\text{s}$ is -
$\left[\sin \right(1.2)=0.93,\text{sin}(24)=-0.9]$

A car is going at a speed of 21.6 km/hr when it encounters a 12.8 m long slope of angle ${30}^{\circ }$ (figure 6-E5).
The friction coefficient between the road and the tyre is \frac{1}{2\sqrt{3}}. Show that no matter how hard the driver applies the brakes, the car will reach the bottom with a speed greater than 36 km/hr. Take g = $10m/s^2$

Find the tension (in $N$) acting on $2m$ mass when the system is released from rest. Assume that the pulleys are ideal and strings are inextensible.

One of the assumptions of kinetic theory of gases is that there is no force of attraction between the molecules of a gas.

State and explain the evidence that shows that the assumption is not applicable for real gases.

A target is made of two plates, one of wood and the other of iron. The thickness of the wooden plate is 4 cm and that of iron plate is 2 cm. A bullet fired goes through the wood first and then penetrates 1 cm into iron. A similar bullet fired with the same velocity from opposite direction goes through iron first and then penetrates 2 cm into wood. If $a_1$ and $a_2$ be the retardations offered to the bullet by wood and iron plates respectively then

Two particles and emitted by a radioactive source at made tracks in a cold chamber as illustrated in the figure. A magnetic field acted downward into the paper. Careful measurements showed that both tracks were circular, the radius of Y track being half that of the Z track. Which one of the following statements is certainly true?

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends ${90}^{\circ }$ angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical. Ratio of $\frac{d_1}{d_2}$ is

${\int }_{-\pi /2}^{\pi /2}cosxdx$

For any two vectors a and b

According to the kinetic theory of gases, between two successive collisions of a gas molecule in an ideal gas travels

Two equal masses (m) are projected at the same angle $\left(\theta \right)$ from two points separated by their range with equal velocities (v). The total momentum at the point of their collision is

A body covers one-third of its journey with speed $u$, next one-third with speed $v$ and the last one-third with speed $w$. Calculate the average speed of the body over the entire journey.

The difference between the apparent frequency of a source of sound as perceived by an observer during its approach and recession is 2% of the natural frequency of the source. If the velocity of sound in air is 300 m/sec, the velocity of the source is (It is given that velocity of source << velocity of sound)

The bulk modulus of $n$ spherical objects is $B$. If it is subjected to uniform pressure $P$, the fractional decrease in radius is -

A circular wooden loop of mass $m$ and radius $R$ rests flat on a horizontal frictionless surface. A bullet, also of mass $m$, and moving with a velocity $V$, strikes the loop at the bottommost point and gets embedded in it. The thickness of the loop is much smaller than $R$. The angular velocity with which the system rotates just after the bullet strikes the loop is

The ionisation energy of 10 times ionised sodium atom (given the ionisation energy of H atom is 13.6eV) is

A composite slab consists of two slabs A and B of different materials but of the same thickness placed one on top of the other. The thermal conductivities of A and B are $k_{1}and{k}_2$ respectively. A steady temperature difference of ${12}^{\circ }C$ is maintained across the composite slab. If $k_1=\frac{k_2}{2}$, the temperature

A wire of negligible mass and length l is stretched between two fixed points P and Q. The wire suffers from a tension T when a mass m is attached to the wire. Distance of this mass from fix point P is a and from Q is b. The attachment of mass to the stretched wire sets it into oscillations of small amplitude. Since the amplitude is small, the transverse movement of R is small so the change in length of the wire can be ignored. Thus tension in the wire can be taken as practically same as T itself. The time period of linear vibratory motion of the mass can be given by $T=\frac{2\pi }{\omega }$, where $\omega$ is a constant of SHM.

Force exerted while trying to bring back the mass attached to the wire from the pulled distance x is given by

Moment of inertia of ring about any of diameters is $I_0$. The moment of inertia of the ring about any tangent perpendicular to its plane is

In the given figure a straight long wire carrying a current i and a rod of length L coplanar with the wire and perpendicular to it moves with a constant velocity v in a direction parallel to the wire. The distance of the wire from the centre of the rod is L. Find the motional emf induced in the rod.

100 mA current gives a full-scale deflection in a galvanometer of $2\Omega$ . The resistance connected with the galvanometer to convert it into a voltmeter to measure $5V$ is

Current density in a cylindrical wire of radius $R$ is given as
$J=\{\begin{array}{c}{J}_0(\frac{x}{R}-1)\text{for}0\le x<\frac{R}{2}\\ J_0\frac{x}{R}\text{for}\frac{R}{2}\le x\le R\end{array}$.
The current flowing in the wire is

A massless uniform rod is subjected to force $F$ at its free end as shown in figure.

The ratio of tensile stress at plane $P_1$ to stress at $P_2$ is

A wire has a mass$(0.3\pm 0.003)g$, radius $(0.5\pm 0.005)mm$ and length $(6\pm 0.06)cm$. The maximum percentage error in the measurement of its density is

In a gravity free space, a particle is connected with four springs as shown below. If the particle is pulled down gently and left, then find the period of oscillation of the particle.
$(\text{Given}:\theta ={45}^{\circ },\beta ={30}^{\circ })$

Four identical point masses, each of mass m and carrying charge +q are placed at the corners of a square of sides 'a' on a frictionless plain surface. If the particles are released simultaneously, the kinetic energy of the system when they are infinitely far apart is

Three sound waves of equal amplitudes have frequencies (v - 1), (v) and (v + 1). They superpose to give beats. The number of beats produced per second will be

A bomb of mass $16\text{kg}$ at rest explodes into two pieces of masses $4\text{kg}$ and $12\text{kg}$ . The velocity of the $12\text{kg}$ mass is $4{\text{ms}}^{-1}$. The kinetic energy of the other mass is

Two rings of same radius and mass are placed such that their centres are at a common point and their planes are perpendicular to each other. The moment of inertia of the system about an axis passing through the centre and perpendicular to the plane of one of the rings is
(Mass of each ring $=m$, radius $=r$)

The density $\rho$ of a piece of metal of mass ${}^{\prime }{m}^{\prime }$ and volume $V$ is given by the relation, $\rho =\frac{m}{V}$. If $m=(275.35\pm 0.01)g$ and $V=(36.41\pm 0.01)c{m}^3$, then the percentage error in $\rho$ will be:

A 5 kg block kept on a frictionless horizontal plane is being acted upon by a 10 N force as shown. Find the acceleration of the block.

The same mass of copper is drawn into two wires $A$ and $B$ of radii $r$ and $3r$ respectively. They are connected in series, and electric current is passed. The ratio of the heat proiduced in $A$ and $B$ is :