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The velocity-time graph is given as





The most probable acceleration-time graph is

The rate at which a particular decay process occurs in a radioactive sample, is proportional to the number of radioactive nuclei present. If N is the number of radioactive nuclei present at some instant, the rate of change of N is $\frac{dN}{dt}=-\lambda N$.



At $t=\infty$, which of the following incorrect?

A sound wave has a wavelength $\lambda$. The distance between the first compression and the fourth rarefaction is

A block of weight $W$ is kept on an inclined surface whose angle of inclination is $\theta$ with horizontal. The component of weight perpendicular to the incline is

A stationary body explodes into three identical fragments such that two of them fly off mutually perpendicular to each other, each with same kinetic energy, $E_o$. The energy of the explosion is

A $10\text{kg}$ mass and a $0.5\text{kg}$ mass hang at two ends of a string that passes over a smooth tube as shown in the figure. The $0.5\text{kg}$ mass moves on a circular path in a horizontal plane about vertical axis. The length of string connecting the $0.5\text{kg}$ mass to the top of the tube is $4\text{m}$ and it makes ${30}^{\circ }$ with the vertical. Then what should be the frequency of revolution (in $\text{cycles/sec}$) of $0.5\text{kg}$ mass so that $10\text{kg}$ mass remains stationary ?

A uniformly charged sphere, entirely made up of charges in vacuum, has a total charge 20 C and radius 10 mm. With reference to infinity, what is the self-potential energy of the system?

Figure shows a loop track of radius $r$. A box starts sliding from a platform at a distance ($h$) above the top of the loop and goes around the loop without falling off the track. Find the minimum value of $h$ for a successful looping. Friction is negligible on all surfaces.

What will be the diameter (in $\text{mm}$) of a water droplet, the pressure inside which is $0.05{\text{N/cm}}^2$ greater than the outside pressure? (Take surface tension as $0.075\text{N/m}$).

In a standing wave pattern in a vibrating air column, nodes are formed at a distance of 4.0 cm. If the speed of sound in air is $328m{s}^{-1}$, what is the frequency of the source?

The Young's modulus of a wire is Y. If the energy per unit volume is E, then the strain will be

Label the eye parts correctly.i) Aqueous humour ii) vitreous humour iii) cornea iv) iris v) pupil

On an inclined plane, two blocks are moving with constant velocities as shown in figure. Neglect the effect of gravity. Find the velocity of their $COM$.

A ring and a disc of same mass and radius roll without slipping along a horizontal surface with the same velocity. If the $K.E.$ of the ring is $8\text{J}$, then $K.E.$ of the disc is

The heat rejected per cycle by a heat engine to the surroundings is $60\%$ of the heat supplied. Find the efficiency of heat engine.

Three particles each of mass 'm', are placed at the corners of an equilateral triangle of side 'a', as shown in Figure. The position vector of the centre of mass is

A cotton reel of mass $m$ and moment of inertia $2{\text{kgm}}^2$ is kept at rest on a smooth horizontal surface. The reel has inner radius $0.2\text{m}$ and outer radius $0.5\text{m}$. A horizontal force $F=200\text{N}$ starts acting as shown is the figure. Find the angular acceleration of the reel.

If the thrust on a rocket moving with a velocity of $300\text{m/s}$ is $210\text{N}$, the rate of combustion of the fuel is

A particle of mass $3m$ is projected from the ground at some angle with horizontal. The horizontal range is $R$. At the highest point of its path it breaks into two pieces $m$ and $2m$. The smaller mass comes to rest and larger mass finally falls at a distance $x$ from the point of projection, where $x$ is equal to

Two identical glass $({\mu }_g=\frac{3}{2})$ equiconvex lenses of focal length f are kept in contact. The space between the two lenses is filled with water $({\mu }_w=\frac{4}{3})$ . The focal length of the combination is

A photoelectric cell is illuminated by a point source of light 1 m away. When the source is shifted to 2 m then

A hollow wooden cylinder of height $h$, inner radius $R$ and outer radius $2R$ is placed in a cylindrical container of radius $3R$. When water is poured into the container, the minimum height $H$ of the container for which cylinder can float inside freely is

Three astronauts, propelled by jet backpacks, push and guide a 2kg asteroid toward a processing dock, exerting the forces shown in figure, with $F_1=4N,F_2=4N,F_3=4N,{\theta }_1={30}^{\circ }and{\theta }_3={60}^{\circ }$. What is the asteroid's acceleration in unit-vector notation ?

A ball is dropped from the top of a tower $120\text{m}$ high and simultaneously another ball is projected vertically upwards from the ground with a speed of $30\text{m/s}$. The two balls will meet after

Momentum of a photon of wavelength $\lambda$ is

Find the maximum value of $F$ such that the system is moving together. Take $g=10{\text{m/s}}^2$.

A body of mass $m=2\text{kg}$, initially at rest and sliding along a frictionless track from a height $\text{h}$ (as shown in the figure) just completes a vertical circle of diameter $\text{AB= 3 m}$. The height $\text{h}$ is equal to

An aircraft moving with a speed of $540\text{km/h}$ is at a height of $6000\text{m}$, just overhead of an anti aircraft gun. If the muzzle velocity of the bullet is $300\text{m/s}$, the firing angle $Q$ from the horizontal for the bullet to hit the aircraft should be

One end of a thin uniform rod of length $L$ and mass $M_1$ is rivetted to the centre of a uniform circular disc of radius $r$ and mass $M_2$, so that both are coplanar. The distance of centre of mass of the combination from the centre of the disc is

A body is projected horizontally from the top of a tower with an initial velocity of $18{\text{ms}}^{-1}$. It hits the ground at an angle of ${45}^{\circ }$ with horizontal. What is the vertical component of velocity when the body strikes the ground?

The energy radiated by a black body at $2300\text{K}$ is found to have maximum intensity at wavelength $1260\text{nm}$, its emissive power being $8000{\text{W/m}}^2$. When the body is cooled to a temperature $T\text{K}$, the emissive power is found to decrease to $500{\text{W/m}}^2$. Find the wavelength at which intensity of emission is maximum at the temperature $T$.

A metal ball falls from a height of $1\text{m}$ on a steel plate and jumps upto a height of $81\text{cm}$. Find the coefficient of restitution for the collision.

A $100\Omega$ resistance and a capacitor of $100\Omega$ reactance are connected in series across a $220V$ source. When the capacitor is $50\%$ charged, the peak value of the displacement current is

A position dependent force $F=7-2x+3x^2$ acts on a small body of mass 2 kg and displaces it from x = 0 to x = 5m. The work done in joules is

A disk of mass $m$ and radius $r$ rotates about an axis passing through its center and perpendicular to its plane with angular velocity $\omega$. Find the percentage change in the kinetic energy when an identical disc is placed over the first disc and both the discs rotate about an axis passing through their center and perpendicular to the plane with same angular velocity $\omega$. Assume there is no friction between the surfaces.

A Force $\overrightarrow{F}=(6\hat{i}+4\hat{j}-2\hat{k})\text{N}$ is applied on a block which is moving with velocity $\overrightarrow{v}=(8\hat{i}+12\hat{j}-3\hat{k})\text{m/s}$. The instantaneous power applied to the particle is

A bucket full of hot water cools from ${75}^{\circ }\text{C}$ to ${70}^{\circ }\text{C}$ in time $t_1$, from ${70}^{\circ }\text{C}$ to ${65}^{\circ }\text{C}$ in time $t_2$ and from ${65}^{\circ }\text{C}$ to ${60}^{\circ }\text{C}$ in time $t_3$, then

A disc of mass $2\text{kg}$ and radius $0.5\text{m}$ is doing translational and rotational motion on a surface as shown in the figure. If $v_{com}=4\text{m/s}$ and $\omega =1\text{rad/s}$ then, find the total kinetic energy of the disc.

In an electromagnetic wave, the amplitude of electric field is 1 V/m. the frequency of wave is $5\times {10}^{14}$Hz. The wave is propagating along z-axis. The average energy density of electric field, in $Joule{m}^3$, will be

One end of a spring of force constant k is fixed to a vertical wall and the other to a block of mass m resting on a smooth horizontal surface. There is another wall at a distance $x_0$ from the block. The spring is then compressed by $2x_0$ and released. The time taken to strike the wall is

A current carrying loop is placed in a uniform magnetic field in four different orientation:I,II, III & IV. Arrange them in the decreasing order of their potential Energy.

A boat which has a speed of $5km/hr$ in still water crosses a river of width $1km$ along the shortest possible path in $20min$. The velocity of the river water in $km/hr$ is

What is the dimensions of $\frac{A}{B}$ in the relation $F=A\sqrt{x}+B{t}^2$, where $F$ is the force, $x$ is the distance and $t$ is the time?

A glass capillary tube of internal radius $r=0.25\text{mm}$ is immersed in water. The top end of the tube projects by $2\text{cm}$ above the surface of water. At what angle does the liquid meet the ends of the tube?

[surface tension of water = $0.07\text{N}{\text{m}}^{-1}$]

The formation of mirage is a consequence of :

If during an adiabatic process, the pressure of a gas is found to be proportional to the square root of the $5^{\text{th}}$ power of its absolute temperature, the ratio $C_p/C_v$ for the gas is

In the figure the pulley $P_1$ is fixed and the pulley $P_2$ is movable. If $W_1=W_2=100N$, what is the angle $\angle A{P}_2{P}_1$? The pulleys are frictionless.

A bullet of mass $10\times {10}^{-3}\text{kg}$ and temperature $0^{\circ }\text{C}$, moving with a speed of $20m{s}^{-1}$ hits an ice block $\left(0^{\circ }\text{C}\right)$ of mass $990\text{g}$ kept at rest on a frictionless floor and gets embedded in it. If ice takes $50\%$ of K.E lost by the system as the heat energy, the amount of ice melted (in $\text{grams}$) approximately is :

(Latent heat of ice $=80\text{cal/g}$)