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Two particles of mass $1\text{kg}$ and $3\text{kg}$ move towards each other under their mutual force of attraction. No other forces act on them. When the velocity of approach of the two particles is $2\text{m/s}$, their centre of mass has a velocity of $0.5\text{m/s}$. When the velocity of approach becomes $3\text{m/s}$, the velocity of the centre of mass of the system of particles is

As shown in the figure, a dust particle with mass $m=5.0\times {10}^{-9}kg$ and charge $q_0=2.0nC$ starts from the rest at point $‘a^{\prime }$ and moves in a straight line to point $b$. What is its speed $v$ at point $b$?

An ideal reversible heat engine converts one sixth of the heat input into work. If the temperature of the sink is reduced by ${62}^{\circ }C$, its efficiency is doubled. Find the temperature of the source and sink respectively (In $\text{Kelvin}$).

The area lying in the first quadrant between $x^2+y^2={\pi }^2$ and $y=\sin x$ and $y-$axis is

A moving coil galvanometer of resistance $100\Omega$ is used as an ammeter using a resistance of $0.1\Omega$. The maximum deflection current in the galvanometer is $0.1\text{mA}$. Find the maximum current that can be measured using the ammeter.

A car moves on a circular road. It describes equal angles about the centre in equal intervals of time. Which of the following statement about the velocity of the car is true

In the arrangement shown in the figure, the coefficient of friction between the two blocks is $\mu =0.5$ and the surface is smooth. The force of friction acting between the two blocks is

A ball of mass $2\text{kg}$ is tied to a string $3\text{m}$ long and is rotated at a constant speed of $30\text{m/s}$ in a vertical circle. The ratio of the tensions at the top and bottom is (Take $g=10{\text{m/s}}^2$)

Two drops of same radius are falling through air with a steady velocity of $5\text{cm/s}$. If the two drops coalesce, the terminal velocity will be

Sara is making icecream. Each batch makes 231 icecreams. If Sara prepares 23 batches, how many icecreams will she have in total?

Two bodies with masses $m_1=2\text{kg}$ and $m_2=4\text{kg}$ have initial velocities $v_1=v_2=5\text{m/s}$ in the directions shown in the figure. If they collide at the origin elastically, then find the position of the center of mass $3\text{seconds}$ after the collision.

The number of turns in the primary and secondary turns of a transformer are $1000$ and $3000$ respectively. If $80V$ AC is supplied to the primary coil of the transformer, then the potential difference per turn of the secondary coil would be

A varying force is given as

$F=\left\{\begin{array}{cc}4\text{N}& 0\le x\le 6\\ 6x\text{N}& 6<x\le 10\\ 3x^2\text{N}& 10<x\le 20\end{array}\right\}$ (where $x$ is meters)

Find the total work done by this force to displaces from $x=4$ to $x=15$ metres.

A ball falling freely from a height of $4.9\text{m}$, hits a horizontal surface. If $e=\frac{3}{4}$, then the ball will hit the surface, second time after

At what angle in degrees should a ray of light be incident on the face of a prism of refracting angle ${60}^o$ so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is $1.524$. Given $si{n}^{-1}\left(0.66\right)={41}^o$, $\sin {19}^o=0.3256$, ${\sin }^{-1}\left(0.4962\right)={30}^o$, $\frac{1}{1.5424}=0.66$

If a particle is moving clockwise in a circle in $x-y$ plane with centre at origin, the direction of angular velocity of the particle is

Force constant of a weightless spring is $16\text{N/m}$. A body of mass $1.0\text{kg}$ suspended from it is pulled down through $5\text{cm}$ from its mean position and then released. The maximum kinetic energy of the body will be

A $0.05\text{m}$ cube has its upper face displaced by $0.2\text{cm}$ by a tangential force of $8\text{N}$. Calculate the modulus of rigidity of the material of the cube.

A charged capacitor is allowed to discharge through a resistance $2\Omega$ by closing the switch $S$ at the instant $t=0$. At time$t=\ln 2\mu \text{s}$, the reading of the ammeter falls half of its initial value. the resistance of the ammeter equals to

A magnetic flux of $8\times {10}^{-4}\text{Wb}$ is linked with each turn of a $200$ turns coil when there is an electric current of $4\text{A}$ in it. Calculate the self inductance of the coil.

The square loop in the figure has sides of length $20\text{cm}$. It has $5$ turns and carries a current of $2\text{A}$. The normal to the loop is at ${37}^{\circ }$ to a uniform field $\overrightarrow{B}=0.5\hat{j}\text{T}$. Find the magnetic moment.

A man running on a horizontal road at $8\text{km/h}$ finds the rain falling vertically. He increases his speed to $12\text{km/h}$ and finds that the drops make an angle ${30}^{\circ }$ with the vertical. Find the speed of the rain with respect to the road (in $\text{km/hr}$).

The photoelectric threshold wavelength of silver is $3250\times {10}^{-10}m$. The velocity of the electron ejected from a silver surface by ultraviolet light of wavelength $2536\times {10}^{-10}m$ is

A semicircular wire $PQ$, of radius $R=0.5\text{m}$, is moving in a uniform magnetic field of magnitude $2\text{T}$, as shown in the figure. If the velocity of the loop is $6\text{m/s}$, then calculate the emf developed across the ends of the wire $PQ$.

A capacitor of capacitance $2\mu \text{F}$ can withstand a maximum voltage of $10\text{kV}$. Another capacitor of capacitance $5\mu \text{F}$ can withstand a maximum voltage of $7\text{kV}$. If the capacitors are connected in series, the combination can withstand a maximum voltage of