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The angle between vectors $\overrightarrow{A}\times \overrightarrow{B}$ and $\overrightarrow{B}\times \overrightarrow{A}$ is

A moving body of mass m and velocity 3 km/h collides with a body at rest of mass 2m and sticks to it. Now the combined mass starts to move. What will be the combined velocity

A milli-ammeter of range 10mA has a coil of resistance 2$\Omega$ . To use it as an ammeter of range 1A, the required shunt must have a resistance of

A and B are two wave trains shown in the figure, the ratio of intensity of A and B is

In a thermodynamic process on an ideal monoatomic gas, the infinitesimal heat absorbed by the gas is given by $T\Delta X$, where $T$ is temperature of the system and $\Delta X$ is the infinitesimal change in a thermodynamic quantity $X$ of the system. For a mole of monoatomic ideal gas $X=\frac{3}{2}\text{R ln}\left(\frac{T}{T_A}\right)+\text{R ln}\left(\frac{V}{V_A}\right)$. Here, $V$ is volume of gas, $T_A$ and $V_A$ are constants. The List - I below gives some quantities involved in a process and List - II gives some possible values of these quantities.
$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(i) Work done by the system in process}1\to 2\to 3& \left(P\right)\frac{1}{3}R{T}_0\text{ln}2\\ \text{(ii)Change in internal energy in process}1\to 2\to 3& \left(Q\right)\frac{1}{3}R{T}_0\\ \text{(iii)Heat absorbed by the system in process}1\to 2\to 3& \left(R\right)R{T}_0\\ \text{(iv) Heat absorbed by the system in process}1\to 2& \left(S\right)\frac{4}{3}R{T}_0\\ & \left(T\right)\frac{1}{3}R{T}_0(3+\text{ln}2)\\ & \left(U\right)\frac{5}{6}R{T}_0\end{array}$

If the process carried out on one mole of monoatomic ideal gas is as shown in figure in the PV diagram with $P_0{V}_0=\frac{1}{3}R{T}_0$. The correct match is-

A uniform chain of mass '$m$' is placed over half ring. If the length of the chain is $\pi R$, then find potential energy of the chain.

A radio station operating at a frequency 100 KHz has two vertical dipole antennas spaced 400 m apart
oscillating in phase. In which directions is the intensity greatest?

In the circuit shown $k_1$ is closed for a long time. Then $k_1$ is opened and $k_2$ is closed simultaneously. If $R=20\Omega ,L=30mH$ and $C=12\mu F$, then maximum charge on the capacitor will be

3 capacitors are connected as shown in the figure. Then the charge on $C_1$ is

Two conical portions each of height 2R are removed from a cylinder of length 4R and radius R as shown in figure. If density of the material is $\mathrm{\rho ,}$ then moment of inertia of given system about the axis of cylinder is

A sphere is thrown along a rough surface as shown. Mass of the sphere is M and radius is R. If t is total time taken to start pure rolling after thrown, then:

An object and its real image are located at distances $25\text{cm}$ and $40\text{cm}$ respectively from the two principal focii of a convex lens on either side. The linear magnification of the image is nearly equal to

Two identical balls, each having a charge of $2.00\times {10}^{-7}C$ and a mass of 100 g, are suspended from a common point by two insulating strings each 50 cm long. The balls are held at a separation 5.0 cm apart and then released. Find
(a) the electric force on one of the charged balls
(b) the components of the resultant force on it along and perpendicular to the string
(c) the tension in the string
(d) the acceleration of one of the balls. Answers are to be obtained only for the instant just after the release.

Three identical rods of length $1\text{m}$ each, having cross-sectional area of $1{\text{cm}}^2$ each and made of aluminium, copper and steel respectively are maintained at temperatures of ${12}^{\circ }\text{C},4^{\circ }\text{C}$ and ${50}^{\circ }\text{C}$ respectively at their separate ends. Find the temperature of their common junction.
$[K_{Cu}=400\text{W/m-K},K_{Al}=200\text{W/m-K},K_{\text{Steel}}=50\text{W/m-K}]$

The relation between the magnitudes of acceleration of pulley $P_1$ ($a_{p_1}$) and that of pulley $P_3$ ($a_{p_3}$) is

Find the frictional force between the two blocks. Take $(g=10{\text{m/s}}^2)$.

In an experiment on photoelectric effect, the emitter and the collector plates are placed at a separation of 10 cm and are connected through an ammeter without any cell. A magnetic field B exists parallel to the plates. The work function of the emitter is 2.39 eV and the light incident on it has wavelengths between 400 nm and 600 nm. Find the minimum value of B for which the current registered by the ammeter is zero. Neglect any effect of space charge.

Two wires of length $l$, radius $r$ and length $2l$, radius $2r$ respectively having same young's modulus are hung with a weight $mg$ as shown in figure. The net elongation is

Scientists of $ISRO$ want to achieve a velocity of rocket $100\text{m/s}$ after time $t=5\text{sec}$. Velocity of ejected burnt fuel is $20\text{m/s}$ w.r.t the rocket. What should be the rate of consumption of fuel (approximately) so that the rocket will achieve the desired velocity?
Initial mass of rocket is $2000\text{kg}$. [Take $g=10{\text{m/s}}^2$]

The equivalent resistance of a voltmeter with coil resistance $R_c=1k\Omega$ and series resistance of 100 k$\Omega$ is .

A body of mass 10 kg is slipping on a rough horizontal plane and moves with a deceleration of 4.0 m/ $s^2$. What is the magnitude of frictional force?

Let $\left[{\epsilon }_0\right]$ denote the dimensional formula of the permittivity of the vaccum and $\left[{\mu }_0\right]$ denote the permeability of the vacuum. If $M$= mass, $L=$ length, $T=$ time and $I=$ electric current, then

Torricelli's barometer uses mercury of density $13.6\times {10}^3{\text{kg/m}}^3$. If mercury is replaced by wine of density $984{\text{kg/m}}^3$, then determine the height of the wine column for normal atmospheric pressure.

The lengths of spring are $l_1$ and $l_2$ when stretched with forces of $4N$ and $5N$ respectively. Its natural length is

A bob is hanging over a pulley inside a car through a string. The second end of the string is in the hands of a person standing in the car. The car is moving with constant acceleration $a$ directed horizontally as shown. The other end of the string is pulled with constant acceleration $a$ vertically down. Find the tension in the string.

A battery of emf E and internal resistance r is connected to a resistor of resistance $r_1$ and Q joules of heat is produced in a certain time t. When the same battery is connected to another resistor of resistance $r_2$ the same quantity of heat is produced in the same time t the value of r is

Projection of $\overrightarrow{A}=\hat{i}-\hat{j}$ along $\overrightarrow{B}=\hat{k}$ is

If fish are flowing at a rate of 10 fish/$m^{2}s$, and the 10 $m^2$ net is dipped at an angle of ${60}^0$ in their path for 1 second, how many fish do we catch?

Two forces $\overrightarrow{F_1}=\hat{i}-2\hat{j}+2\hat{k}$ $\text{N}$ and $\overrightarrow{F_2}=-3\hat{i}+\hat{j}-3\hat{k}$ $\text{N}$ acts on a body and causes it to displace from position $\overrightarrow{s_1}=\hat{i}-\hat{j}$ $\text{m}$ to $\overrightarrow{s_2}=2\hat{j}-a\hat{k}$ $\text{m}$. Find the value of ${}^{\prime }{a}^{\prime }$ if the amount of net work done by the forces is $5\text{J}$ during the displacement.

The pressure and density of a diatomic gas $(\gamma =\frac{7}{5})$ changes from $(P,\rho )$ to $(P^{\prime },{\rho }^{\prime })$ during an adiabatic change. If $\frac{{\rho }^{\prime }}{\rho }=32$, then the value of $\frac{P^{\prime }}{P}$ is

As shown in figure a person is pulling a mass $m$ from ground on a fixed rough hemispherical surface up to the top of the hemispherical surface with the help of a light inextensible string. The radius of hemisphere is $R$ and friction co-efficient between surfaces is $\mu$. Assume that the block is pulled with negligible velocity, the work done by tension in string is

A square shaped slab of lead with side $50\text{cm}$ and thickness $10.0\text{cm}$ is subjected to a shearing force (on its narrow face) of magnitude $9.0\times {10}^4\text{N}$. The lower edge is riveted to the floor as shown in figure. How much is the upper edge displaced, if the shear modulus of lead is $5.6\times {10}^9\text{Pa}$?

The power radiated by a black body is $P$, and it radiates maximum energy around the wavelength ${\lambda }_0$. If the temperature of the black body is now changed so that it radiates maximum energy around a wavelength $\frac{3{\lambda }_0}{4}$, the power radiated by it will increase by a factor of

A boy standing at the top of a tower of $30\text{m}$ height drops a stone. Assuming $g=10{\text{ms}}^{-2}$ the magnitude of velocity with which it hits the ground is

An ice cube contains a glass ball. The cube is floating on the surface of water contained in a trough. What will happen to the water level, when the cube melts?

A stone is dropped into a river from a bridge 125 m above the water. Another stone is thrown vertically down 1.00 s after the front is dropped. The stones strike the water at the same time.

Which of these is the correct position-time curve of the stones w.r.t. the person who is throwing them?

Two blocks with masses $m_1=3\text{kg}$ and $m_2=4\text{kg}$ are touching each other on a fricionless surface. If a force of $5\text{N}$ is applied on $m_1$ as shown in figure. The acceleration of each block is

A train moves towards a stationary observer with speed 34 m/s. The train sounds a whistle and its frequency registered by the observer is $f_1$. If the train's speed is reduced to 17 m/s, the frequency registered is $f_2$ . If the speed of sound is 340 m/s then the ratio $\frac{f_1}{f_2}$ is :

The current sensitivity of a moving coil galvanometer is 5 divisions per milliampere and voltage sensitivity is 20 divisions per volt find the resistance of the Galvanometer

An edge of a variable cube is increasing at the rate of 3cm/s. How fast the volume of the cube increasing when the edge is 10 cm long?

A particle undergoes a displacement $△\overrightarrow{r}=2.0\hat{i}-3.0\hat{j}+6.0\hat{k}$, ending with the position vector $\overrightarrow{r}=3.0\hat{j}-4.0\hat{k}$, in meters, What was its initial position vector?

What is the energy required to completely remove an electron from the $H{e}^{+}$ atom if it is initially present in $n=3$ level?

Show that the height of the cylinder of greatest volume which can be inscribed in a circular cone of height h and having semi-vertical angle is one-third that of the cone and the greatest volume of the cylinder is $\frac{4}{27}\pi h^{3}ta{n}^2\alpha$.

One litre of a gas weighs 4g at 300 K and 1 atm. If the pressure is reduced to 0.75 atm, the temperature at which one litre of the same gas weights 2g is