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Arrangement of pipes carrying fluids of densities ${\rho }_1,{\rho }_2\&{\rho }_3$ is shown in the figure below. If the pipe carrying fluid of density ${\rho }_1$ makes an angle $\theta$ with the horizontal, then what is the value of ${}^{\prime }{h}^{\prime }$ ?

Three uniform bricks each of length $60\text{cm}$ and mass $m$ are arranged as shown in the figure. Considering only the length of the bricks, find the distance of centre of mass of the system of bricks from the wall.

The horizontal distance x and the vertical height y of a projectile at a time t are given by

$x=atandy=b{t}^2+ct$

where a, b and c are constants. What is the magnitude of the velocity of the projectile 1 second after it is fired?

A hockey player receives a corner shot at a speed of $15\text{m/s}$ at an angle ${30}^{\circ }$ with $y$ - axis and then kicks the ball along $x$ - axis with speed $30\text{m/s}$. If the mass of the ball is $150\text{gm}$ and it remains in contact with the hockey stick for $0.01\text{s}$, then find the force exerted on the ball along $x$ - axis.

The vector sum of the forces $10N$ and $6N$ can be

${10}^{23}$ molecules of nitrogen, argon and carbon dioxide each are confined in three closed boxes of a large fixed volume V, at the same temperature T. Knowing the molecular weights of the gases are in the order $M_{C{O}_2}$>$M_{N_2}$>$M_{A_r}$, what can you say about the pressures of the gases in each container?

A vector $\overrightarrow{F_1}$ is along $x$-axis. If its vector product with another vector $\overrightarrow{F_2}$ is zero, then $\overrightarrow{F_2}$ could be

A gas mixture consists of 2 moles of oxygen and 4 moles of Argon at temperature $T$. Neglecting all vibrational degrees of freedom, the total internal energy of the system is

Find the derivative of functions $cosx$, $cosecx$, $secx$, $cotx$ using first principle​​​​​​

A pizza shaped wire carries a current of 5A as shown below. If the angle subtended is $\frac{\pi }{3}$ radians and its radius is 10 cm, what will be the strength of magnetic field due to it at its centre? (Magnetic field due to straight wire at any point passing through it will be zero)

The rms speed of oxygen molecules in a gas is v. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become

Density of ice is $\rho$ and that of water is $\sigma$. What will be the decrease in volume when a mass M of ice melts

A rectangular loop ABCD, carrying a current 15 A, is placed near a long wire carrying a current 40 A. The resultant force acting on the loop is

The inward and outward electric flux for a closed surface in units of $N-m^2/c$ are respectively $8\times {10}^3$ and $4\times {10}^3$. Then the total charge inside the surface is [where $ϵ$ = permittivity constant]

Find velocity of block $B$ at the instant shown in figure. Assume the surface and pulleys to be frictionless and string to be inextensible.

If the speed of the particle is $v=8t^3-3t^2+2\text{m/s}$, the distance of the particle as a function of time $t$ and the average speed in $2\text{sec}$ respectively are

A heavy but uniform rope of length $L$ is suspended from a ceiling. A particle is dropped from the ceiling. At the same instant, the bottom end is given a jerk. Find the velocity of wave pulse at an instant when particle and pulse are at same level.

if f(x)=x cos x, find f"(x), or $\frac{d^{2}y}{d{x}^2}$

A particle is projected at an angle with the horizontal such that it follows a trajectory given by the equation $y=6x–2x^2.$ Find the maximum height attained by it.

Two equal forces $\overrightarrow{F}$ are acting such that the magnitude of their resultant is equal to F. They must be inclined at an angle of __ degrees

A watermelon seed has the following coordinates: x=-5.0 m, y=9.0 m and z=0 m. Find its position vector



(a) In unit-vector notation and as

(b) A magnitude and

(c) An angle relative to the positive direction of the x-axis

While moving upwards in the lift, Nishant rolls a coin that covers $2\text{m}$ before coming to rest. How much does the coin travels with respect to Nishant from lift's frame of reference ?

An infinitely long rod lies along the axis of a concave mirror of focal length f. The near end of the rod is at a distance u > f from the mirror. Its image will have a length

A bullet with mass $200\text{gm}$ and velocity $v_b\text{m/s}$hits a ballistic pendulum bob of mass $800\text{gm}$ and rises along with pendulum to a height $h=0.2\text{m}$. Determine the value of $v_b.$ Take $g=10{\text{m/s}}^2$.

A stone is projected from the point on the ground in such a direction so as to hit a bird on the top of a telegraph post of height $h$ and then attain the maximum height $\frac{3h}{2}$ above the ground. If at the instant of projection, the bird were to fly away horizontally with uniform speed, find the ratio between horizontal velocities of the bird and stone if the stone still hits the bird while descending.

In a vernier calipers $10$ smallest divisions of the vernier scale are equal to $9$ smallest divisions of the main scale. If the smallest division of the main scale is half millimeter, then the vernier constant is:

All the surfaces shown in the figure are frictionless. The mass of the inclined wedge is $M$, that of the block is $m$ and spring constant of the spring is $k$. Initially, the block and wedge are at rest. The spring is stretched by a small length and then released. Find the time period of oscillation of the system.

In the figure, $n$ moles of a monoatomic ideal gas undergo the process $ABC$ as shown in the $P-V$ diagram. The process $AB$ is isothermal and $BC$ is isochoric. The temperature of the gas at $A$ is $T_0.$ Total heat given to the gas during the process $ABC$ is measured to be $Q$.





Heat absorbed by the gas in the process $BC$ is

What will be the reading of spring balance shown in the figure, if it is moving in upward direction with a constant acceleration $a=2{\text{m/s}}^2$ and an initial velocity $v=2\text{m/s}$. (Take $g=10{\text{m/s}}^2$)

A container has two layers of immiscible liquids $1$ and $2$ with refractive indices $1.5$ and $4/3$ respectively. What is apparent position of an object placed at the bottom of the vessel. Thickness of the layers are $3\text{cm}$ and $8\text{cm}$ respectively?

A ball of mass $m=2\text{kg}$ is projected at an angle $\theta ={45}^{\circ }$ with initial velocity of $u=10\text{m/s}$ as shown in figure. Find the torque acting on the particle due to gravity about origin at it's maximum height.

(Take $g=10m/s^2$)

The distance of a point from the earth's centre where the resultant gravitational field due to the earth and the moon is zero is...( The mass of the earth is 6.0$\times$ ${10}^{24}$ kg and that of the moon is 7.4$\times$ ${10}^{22}$ kg. The distance between the earth and the moon is 40 $\times$ ${10}^5$ km.

An unlabelled analog clock is kept in front of the mirror. Actual time on the clock is $4\text{hr}20\text{minutes}15\text{sec}$. What will be the time in the mirror image of the same clock?

At distance $r$ from a point charge, the ratio $\frac{u}{v^2}$ (where $u$ is energy density and $v$ is potential) is best represented by

Two racing cars of masses $m_{1}and{m}_2$ are moving in circles of radii $r_{1}and{r}_2$ respectively. Their speeds are such that each makes a complete circle in the same duration of time t. The ratio of the angular speed of the first to the second car is

Find the work done in lifting $5$ cement bags, each of mass $25\text{kg}$ to the top of a building of height $50\text{m}$.(consider $g=10m/s^2$)

Three concentric spheres are shown in figure. Find the potential on surface of shell $B$. Charge on $A$, $B$ and $C$ are $3Q,-2Q$ and $Q$ respectively.

The condition such that the potential difference, $V_A-V_B=0$ is.

Two forces whose magnitudes are in the ratio $3:5$ give a resultant of $28\text{N}$. If the angle of their inclination is ${60}^{\circ }$, find the magnitude of the larger force.

If the length of a simple pendulum is recorded as $(90.0\pm 0.02)\text{cm}$ and period as $(1.9\pm 0.02)\text{s}$, the percentage of error in the measurement of acceleration due to gravity is

In the given detector circuit, the suitable value of carrier frequency is

A spherical planet far out in space has a mass $M_0$ and diameter $D_0$. A particle of mass m falling freely near the surface of this planet will experience an acceleration due to gravity which is equal to

A man is capable of swimming at $10\text{m/s}$ in still water. If he has to cross a river flowing at $5\text{m/s}$ and reach the opposite shore in the least possible time, how far along the river will he drift if distance between the two shore is $100\text{m}$.

The work done in blowing a soap bubble of $10\text{cm}$ radius is -

(surface tension of soap $=0.03\text{N/m}$).

The velocity of water in a river is $18{\text{kmh}}^{-1}$ near the surface. If the river is $5\text{m}$ deep, then the shearing stress between the surface layer and the bottom layer is:

(Given - coefficient of viscosity of water, $\eta ={10}^{-3}\text{Pa.s}$)