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A body starts from rest, and moves with an acceleration of $1m{s}^{-2}$. The displacement (in $\text{m}$) of the body in $5$ seconds is

A wire is $4\text{m}$ long and has a mass of $0.2\text{kg}$. The wire is kept horizontally. A transverse pulse is generated by plucking one end of the taut (tight) wire. The pulse makes four trips back and forth along the
cord in $0.8\text{sec}$. The tension is the cord will be

Find the derivative of $y=\frac{(t^2-1)}{(t^2+1)}$.

An RC circuit consists of a resistance R = 5 MΩ and a capacitance C = 1.0 μF connected in series with a battery. In how much time will the potential diffrence across the capacitor become 8 times that across the resistor? (Given $lo{g}_e\left(3\right)=1.1$ )

(IIT-JEE 2005)

Find net capacitance between A & B

A motor cyclist moving with a velocity of $72\text{km/h}$ on a flat road takes a turn on the road at a point where the radius of curvature of the road is $20\text{m}$. The acceleration due to gravity is $10{\text{ms}}^{-2}$. In order to avoid skidding, he must not bend with respect to the vertical plane by an angle greater than

A swimmer crosses the river along the line making an angle of 45° with the direction of flow. velocity of the river water is 5 m/sec Swimmer takes 6 seconds to cross the river of width 60 metre find the velocity of the swimmer with respect to water .

A star initially has ${10}^{40}$ deuterons. It produces energy via the processes,



${}_1{\text{H}}^2{+}_1{\text{H}}^2\to {}_1{\text{H}}^3+p$



${}_1{\text{H}}^2{+}_1{\text{H}}^3\to {}_2{\text{He}}^4+n$



The masses of the nuclei are as follows,



$m{(}_1{\text{H}}^2)=2.014\text{amu};m{(}_1{\text{H}}^3)=3.016\text{amu}m\left(p\right)=1.007\text{amu};m\left(n\right)=1.008\text{amu}m{(}_2{\text{He}}^4)=4.001\text{amu}$



If the average power radiated by the star is ${10}^{16}\text{W}$, the deuteron supply of the star is exhausted in a time period of the order of

[Take $1\text{amu}=931.5\frac{\text{MeV}}{c^2}$ ]

A charged particle with charge to mass ratio $(\frac{q}{m}=\frac{{10}^3}{19}{\text{C kg}}^{-1})$ enters a uniform magnetic field $\overrightarrow{B}=20\hat{i}+30\hat{j}$$+50\hat{k}\text{T}$ at time $t=0$ with velocity $\overrightarrow{v}=20\hat{i}+50\hat{j}+30\hat{k}\text{m/s}$. Assume that magnetic field exists in large space. The pitch of the helical path of the motion of the particle approximately will be

A uniform circular disc of mass 200 g and radius 4.0 cm is rotated about one of its diameter at an angular speed of 20 rad/s. Its angular momentum about the axis of rotation is (in kg-$m^2$/s)

Two similar rods are joined as shown in figure. Then temperature of junction is (assume no heat loss through lateral surface of rod and temperatures at the ends are shown in steady state)

The gravitational field due to a mass distribution is given by $E=\frac{K}{r^3}$ in $\text{x-}$direction. Taking the gravitational potential to be zero at infinity, find its value at a distance $x$

A monatomic ideal gas follows the process $T{V}^{3/2}=$ constant. If the molar heat capacity for this process is $\left(\frac{5R}{y}\right)$, then the value of $y$ is

A particle of charge $-q$ and mass $m$ moves in a circular orbit of radius $r$ about a fixed charge $+Q$. The relation between the radius of the orbit $r$ and the number of revolution per second $n$ is

Light with energy flux of $18{\text{W/cm}}^2$ falls on a non-reflecting surface, of area $20{\text{cm}}^2$, at normal incidence. The momentum delivered in $30\text{minute}$ is -

When a surface is illuminated with light of wavelength , the stopping potential is V. When the same surface is illuminated by light of wavelength 2 $\lambda$, the stopping potential is $\frac{V}{3}$. Threshold wavelength for metallic surface is:

Imagine a hypothetical fluid made only of helium nuclei (practically, just liquid ionized helium), and let's say, it's seen that it floats on water, like oil. A similar fluid made of uranium${(}_{92}^{238}U)$ nuclei, what will happen to it once put in the same water?

The spring shown in figure is kept I a stretched position with extension $x_0$ when the system is released. Assuming the horizontal surface to be frictionless, find the frequency of oscillation.

A block of density $2000kg/m^3$ and mass 10 kg is suspended by a spring stiffness 100 N/m. The other end of the spring is attached to a fixed support. The block is completely submerged in a liquid of density $1000kg/m^3$. If the block is in equilibrium position then,

Equal volume of two immiscible liquids of densities $\rho$ and $2\rho$ are filled in a vessel as shown in the figure. Two small holes are punched at depths h/2 and 3h/2 from the surface of lighter liquid. If $v_1$ and $v_2$ are the velocities of efflux at these two holes, then $v_1/v_2$ is

A small block of mass $m$ is placed over a long plank of mass $M$. Coefficient of friction between them is $\mu$. Ground is smooth. At $t=0$, $m$ is given a velocity $v_1$ and $M$ a velocity $v_2(>v_1)$ as shown in the figure. After this $M$ is maintained at constant acceleration $a(<\mu g)$.



The variation of velocity of block as a function of time is shown as

As shown in the figure, sand is falling from a stationary hopper onto a freight car which is moving with a uniform velocity $v_0=10\text{m/s}.$ The sand falls at the rate of $\mu =5\text{kg/s}.$ How much force is needed to keep the freight car moving at the speed $v_0$?

For the shown $L-C$ circuit, the maximum value of charge on capacitor is $200\mu \text{C}$. When the charge become $100\mu \text{C}$ what is the value of $\left|\frac{di}{dt}\right|$, where $i$ is the instantaneous current and $t$ is the time ? [Note that the capacitor is first charged and then connected with the inductor ]

A curve has a radius of $50\text{m}$ and a banking angle ($\theta$) of ${15}^{\circ }$. What is the maximum safe speed for a car on this curve if the road does not offer any friction? Take $g=10{\text{m/s}}^2$.

For three different metals A,B and C, the photo-emission is observed when exposed to radiation [${\nu }_{oA}<{\nu }_{oB}<{\nu }_{oC}$]. The graph of maximum kinetic energy versus the incident frequency should be represented as: