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The area of cross-section of the two arms of a hydraulic press is $1{\text{cm}}^2$ and $10{\text{cm}}^2$, respectively as shown in figure. A force of $10\text{N}$ is applied on the water in the thinner arm. What force should be applied on the water in the thicker arms so that the water may remain in equilibrium?

A car is moving in a circular horizontal track with a constant speed. The centripetal force acting on the car is $75\text{N}$ . If the car’s speed is doubled, what is the new centripetal force required for the car to drive on the same circular track?

The orbital velocity of an artificial satellite in a circular orbit just above the earth’s surface is $v$. For a satellite orbiting at an altitude of half of the earth’s radius, the orbital velocity is

Find the values of x for which A $si{n}^{2}x$ is minimum and Maximum. A is some positive constant.

A $10\text{kg}$ mass and a $0.5\text{kg}$ mass hang at two ends of a string that passes over a smooth tube, as shown in the figure. The $0.5\text{kg}$ mass moves on a circular path in a horizontal plane about a vertical axis. The length of string connecting the $0.5\text{kg}$ mass to the top of the tube is $4\text{m}$ and it makes ${30}^{\circ }$ with the vertical. Then what should be the frequency of revolution (in $\text{cycles/sec}$) of $0.5\text{kg}$ mass so that $10\text{kg}$ mass remains stationary ? (Take $g=10{\text{m/s}}^2)$

Three capacitors having capacitance $20\mu \text{F},30\mu \text{F}$ and $40\mu \text{F}$ are connected in series with a $12\text{V}$ battery. The amount of work done by the battery in charging the combination will be:

Two particles are projected simultaneously with the same speed $v$ in the same vertical plane with angles of elevation $\theta$ and $2\theta$, where $\theta <{45}^{\circ }$. At what time will their velocities be parallel ?

An inductance $L$ and a resistance $R$ are first connected to a battery. After attaining steady state, the battery is disconnected but $L$ and $R$ remain connected in a closed circuit. Then, the current reduces to $37\%$ of its initial value in time -

$\int \frac{si{n}^{2}x}{1+cosx}$

A total of Q charge flows from resistance R in time interval T such that current vs time graph is obtained as shown in figure. The heat generated in resistance is:

A ray of light passes from vacuum into a medium of refractive index $\mu$,the angle of incidence is found to be twice the angle of refraction. Then the angle of incidence is

Twelve wires, each having equal resistance of 2 Ω, are joined to form a cube. The equivalent resistance between diagonally opposite points is .

A spherical black body has a radius $R$ and steady surface temperature $T$, heat sources in it ensure the heat evolution at a constant rate and distributed uniformly over its volume.
If the radius is decreased by half, what would be the new temperature of the surface at steady state ?
Assume surrounding to be at absolute zero and heat evolution rate through unit volume remains the same.

A resistance of $20\Omega$ is connected to a source of alternating current rated $220\text{V}$, $50\text{Hz}$. Find the maximum instantaneous current in the resistor.

In the figure shown, the acceleration of A is $2m/s^2$ to left and acceleration of B is 1 $m/s^2$ to left.

Find the magnitude of acceleration of C in $m/s^2$

For a quadratic equation $a{x}^2+bx+c=0$, if sum of the roots is - 1 and product of the roots is 1, then which of the following holds true?

The displacement $x$ of a particle of mass $m\text{kg}$ moving in one dimension is related to the time $t$ by the equation $t=4x+3$ where $x$ is in metre and $t$ is in second. The work done by the force in the first six seconds in joules is

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends ${90}^{\circ }$ angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical.

Ratio $\frac{d_1}{d_2}$ is:

A Carnot heat engine works between the temperatures ${427}^{\circ }\text{C}$ and ${27}^{\circ }\text{C}$. What amount of heat should it consume per second to deliver mechanical work at the rate of $1.0\text{kW}$?

If $V_s$ is the gravitational potential on the surface of the earth, then what is its value at distance ${\left(\frac{1}{4}\right)}^{th}$ of the radius of the earth from the centre of the earth?

(Assume that mass distribution is uniform)

At what point of a projectile motion acceleration and velocity are perpendicular to each other

The given infinite grid consists of hexagonal cells of six resistors each of resistance R. Then $R_{12}$ is equal to

The trajectory of a projectile in a vertical plane is given by $y=ax-b{x}^2,$ where $a$ and $b$ are constants and $x$ and $y$ are respectively horizontal and vertical distances of the projectile from the point of projection. The maximum height attained by the particle and the angle of projection from the horizontal are

The two plates of a parallel plate capacitor of same lengths have unequal areas as $100{\text{cm}}^2$ and $120{\text{cm}}^2$ respectively. If the separation between the plates is $10\text{mm}$ throughout, then its capacitance will be

The motor of an engine is rotating its axis with an angular velocity of $100\text{rev per min}$. It comes to rest in $15\text{s}$ after being switched off, assuming constant angular deceleration. What is the number of revolutions made by it before coming to rest ?

In the given circuit, find the ratio of $i_1$ and $i_2$. $i_1$ is the initial (at $t=0$) current, and $i_2$ is the steady state (at $t=\infty )$ current through the battery:

In the figure shown, the final charge on the capacitors of capacitance $2C$ and $3C$ are respectively.

A police van moving on a highway with a speed of 30 km/h fires a bullet at a thief's car speeding away in the same direction with a speed of 192 km/h. If the nuzzle speed of the bullet is 150 m/s, with what speed does the bullet hit the thief's car?

A small square loop of wire of side l is placed inside a large square loop of side L $(L\gg l)$. The loops are coplanar and their centres coincide. The mutual induction of the system is proportional to

The wavelength of the first member of Balmer series of hydrogen is $6563\times {10}^{-10}\text{m}$. Calculate the wavelength of its second member.

Two metal plates have potential difference of 300V are 0.01m apart. Then the electric field between the plates is

If $E$ is the electric field intensity of an electrostatic field, then the electrostatic energy density is proportional to :

Find the second order
derivatives of the function.

Two coherent light sources S₁ and S₂ (l= 6000 Å) are 1mm apart from each other. The screen is placed at a distance of 25 cm from the sources. The width of the fringes on the screen should be