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Inside a hollow uniform sphere of mass M , a uniform rod of lenght R$\sqrt{2}$ is released from the state of rest. The mass of the rod is same as that of the sphere. If the inner radius of the hollow sphere is R then find out horizontal displacement of sphere with respect to earth in the time in which the rod becomes horizontal.

A metalic metre stick moves with a velocity of 2 m $s^{-1}$ in a direction perpendicular to its length and perpendicular to a uniform magnetic field of magnitude 0.2 T. Find the emf induced between the ends of the stick.

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1. 2.5

A projectile is fired with a speed $u$ at an angle $\theta$ above the horizontal floor. The coefficient of restitution between the projectile and the floor is $e$. Find the position from the starting point, where the projectile will land at its second collision.

The rms value of the current given by, $I=4\sin \left(100\pi t\right)\text{A},$ in a complete cycle, is



$[$0.77 Mark$]$

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A body of mass 5 kg is moving along the x-axis with a velocity 2$m{s}^{-1}$. Another body of mass 10 kg is moving along the y-axis with a velocity $\sqrt{3}m{s}^{-1}$. If they collide at the origin and stick together, then the final velocity of the combined mass is

A bus starts moving with acceleration $2{\text{ms}}^{-2}$. A cyclist $96\text{m}$ behind the bus starts simultaneously towards the bus at a constant speed of $20\text{m/s}$. After what time will he be able to overtake the bus?

The mean free path of conduction electrons in copper is about $4\times {10}^{-8}\text{m}$. For a copper block, find the electric field which can give, on an average, $1\text{eV}$ energy to conduction electrons.

A uniform current carrying ring of mass $m$ and radius $R$ is connected by a massless string as shown. If a uniform magnetic field $B_0$ exists in the region to keep the ring in horizontal position, then the current in the ring is

A solid sphere as shown is rolling without slipping. What is the maximum distance travelled by sphere on the inclined plane ?

Given $g=10{\text{m/s}}^2$

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Vector A has a magnitude of 2 units and makes an angle of $0^{\circ }$ with the positive x- axis, Vector B has a magnitude of 6 units and makes an angle of ${30}^{\circ }$
(clock-wise) with the positive y- axis. Find the magnitude of the resultant of Vector A& B and also the angle that it makes with Vector A.

A circuit contains a resistance $R$ and an inductance $L$ in series. An alternating voltage $V=V_0\sin \omega t$ is applied across it. The currents in $R$ and $L$ respectively will be ($I_0$ is the peak current)

Two wires of diameter 0.25 cm, one made of steel and the other made of brass are loaded as shown in Fig. 9.13. The unloaded length of steel wire is 1.5 m and that of brass wire is 1.0 m. Compute the elongation of the steel and the brass wires.

Consider a cube having elastic modulus E and restrained in all the direction, subjected to increase in temperature $T^{\circ }C$, stress developed at sides of cube will be. $(\alpha$ is the coefficient of thermal expansion and $\mu$ is Poisson's ratio).

The frequency of oscillation of current in the inductor is

The value of $(\overrightarrow{A}+\overrightarrow{B})\times (\overrightarrow{A}-\overrightarrow{B})$ is

An isosceles prism of angle ${120}^{\circ }$ has a refractive index 1.44. Two parallel monochromatic rays enter the prism parallel to each other in air as shown in the figure. The rays emerging from the opposite faces.

A particle moves according to the equation $\frac{dv}{dt}$ = $\alpha$ - $\beta$ v , where $\alpha$ and $\beta$ are constants. Find the velocity as a funtion of time. Assume body starts from rest.

Work done by a conservative force on a system is equal to

A man is rowing a boat with a constant velocity $v_0$ in a river. The contact area of boat with water is $A$, coefficient of viscosity of water is $\eta$ and the depth of river is $D$. The force required to row the boat is given by,

A parallel beam of white light is incident normally on a water film of thickness $1.0\times {10}^{-4}\text{cm}$. Find the wavelength in the visible range $(400\text{nm}-700\text{nm})$ which are strongly transmitted by the film.



Refractive index of water, $\mu =1.33$.

If a cricket player can throw a ball to a maximum horizontal range of $60m$, what is the maximum vertical height (in $m$) to which he can throw?

The potential energy of a particle that is free to move along $x-\text{axis}$ is given by $U=(x^3-\frac{x^2}{2})\text{J}$, where $x$ is in $\text{meter}$. The particle is initially at $x=0$. Find the magnitude of force at $x=2\text{m}$.

The acceleration due to gravity at a height $2\text{km}$ above the earth surface is same as that of acceleration due to gravity at depth ${}^{\prime }{d}^{\prime }$ below the surface of earth. Then d is equal to

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 $\frac{dN}{dt}=-\lambda N$.

Consider radioactive decay of A to B which may further decay either to X or to Y. ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ are decay constant for A to B decay, B to X decay and B to Y decay respectively. If at $t=0$ number of nuclei of A,B,X and Y are $N_0,N_0$, zero and zero respectively and $N_1,N_2,N_3,N_4$ are number of nuclei A,B,X and Y at any instant.



Rate of accumulation of at any instant will be

Suppose, the acceleration due to gravity at the earth's surface is $10{\text{ms}}^{-2}$ and at the surface of Mars is $4.0{\text{ms}}^{-2}$. A $60\text{kg}$ passenger goes from the earth to Mars in a spaceship moving with a constant velocity. Neglect all other objects in space. Which part of the figure best represents the weight (net gravitational force) of the passenger as a function of time?

A motor car is moving with speed 30m/s on a circular path of radius 500m. Its speed is increasing at the rate of $2m/s^2$. What will be its resultant acceleration?