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A particle rests on the top of a hemisphere of radius $R.$ Find the smallest horizontal velocity that must be imparted to the particle if it is to leave the hemisphere without sliding down is

The phase difference between input and output voltages of a CE circuit is

Two insulated charged spheres of radii 20cm and 25cmrespectively and

having an equal charge Q are connected by a copper wire, then they are

separated

A uniform rod of density $\rho$ is placed in a wide tank containing a liquid of density ${\rho }_0$ (${\rho }_0>\rho$). The depth of liquid in the tank is half the length of the rod. The rod is in equilibrium, with its lower end resting on the bottom of the tank. In this position the rod makes an angle $\theta$ with the horizontal, then

A stationary hydrogen atom of mass $m$ in the ground state achieves minimum excitation energy after head-on inelastic collision with a moving hydrogen atom. Find the velocity of moving hydrogen atom before collision.

The radii of a spherical capacitor are equal to a and b(b > a). The space between them is filled with a dielectric of dielectric constant K and resistivity $\rho$. At t = 0, the inner electrode is given a charge $q_0$. The Charge q on the inner electrode as a function of time is given by $q=q_0{e}^{-\frac{t}{N\rho K{\epsilon }_0}};$ then N is___

Find the distance between object $\left(O\right)$ and final image formed after two successive reflections. Take the first reflection on mirror $M_1$ having radius of curvature $20\text{cm}$

Two masses are connected by a string as shown in the figure over a frictionless pulley attached to a fixed wedge. The acceleration of the system is

The physical quantity with same dimensional formula as Planck’s constant h is

A time varying force is represented as $F=2t\text{N}$. Find out the impulse imparted by this force during the first second.

The region between two concentric spheres of radii ‘a’ and ‘b’ respectively (see figure,) has volume charge density r = A/r, where A is a constant and r is the distance from the centre. At the centre of the sphere is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant is

Two bodies M and N of equal masses are suspended from two separate mass less springs of spring constant $k_1$ and $k_2$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of M to that of N is

The X-ray wavelength of $L_{\alpha }$ line of platinum $(Z=78)$ is $1.30\dot{A}$. The X –ray wavelength of $L_{\alpha }$ line of Molybdenum (Z=42) is

Suppose a diatomic gas gets ionised to a certain extent without any expenditure of heat energy. If the fractional change in the number of moles of the gas be $\eta$, then ignoring any energy loss, the fractional change in the temperature of the gas will be:

A particle is fired vertically upwards with a speed of $9.8km/s$. Find the maximum height attained by the particle. Radius of earth = $6400km$ and $g$ at the surface = $9.8{m/s}^2$. Consider only earth’s gravitation.

The ratio of average translational kinetic energy to rotational kinetic energy of a diatomic molecule at temperature T is

Which of the following equations represents a wave?

The susceptibility of annealed iron at saturation is 5500. Find the permeability of annealed iron at saturation

Which of the following is the correct order of wavelength?

A horizontal board of negligible mass supports a mass weighing $20\text{N}$ at a distance of $2\text{m}$ from the point $A$. Calculate the weight which must be placed at point $B$, which is $5\text{m}$ from point $A$, in order to keep the board in rotational equilibrium?

How much faster does a cup of tea cool by $1^{\circ }C$ when at 373K than when at 303K. Consider the tea as a black body, the room temperature being 293K and $\sigma =5.7\times {10}^{-8}W/m^2{K}^4$

A metal bar of length $1\text{m}$ and area of cross-section $1{\text{mm}}^2$ is clamped between two rigid supports. For the material of rod, its Young's modulus is $Y={10}^{11}{\text{N/m}}^2$ and coefficient of linear expansion is $\alpha =1.8\times {10}^{-5}{}^{\circ }{C}^{-1}$. If the temperature of the rod is decreased by $\Delta T={10}^{\circ }C$, the force exerted by the rod on the supports is

An electron makes $3\times {10}^3$ revolutions per second in a circle of radius 0.5 angsfrom. Find the magnetic field B at the centre of the circle.

A body P strikes head-on with another body Q of mass that is 'p' times that of body P and moving with a velocity that is $\frac{1}{q}$ of the velocity of body P. If the body P comes to rest, the coefficient of restitution is

A man climbs a light rope whose breaking strength is $\eta$ times his weight $(\eta <1)$. The maximum acceleration of the man so that the rope just breaks is

A ball of mass $m$ kept at the centre of a long string of length $L$ is pulled from the centre in a perpendicular direction and released. The time period of ball for small oscillations is
($T^{\prime }$ is tension in the string)

The velocity – time graph of a stone thrown vertically upward with an initial velocity of $30{\text{ms}}^{-1}$ is shown in the figure below. The velocity in the upward direction is taken as positive and that in downward direction as negative. What is the maximum height to which the stone rises?

Three concentric spherical metallic shells A, B and C of radii a, b and c (a < b < c) have charge densities of $\sigma$, – $\sigma$ and $\sigma$ respectively, then answer the following questions:

$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ & \text{(Point under consideration)}& & \text{(Electric field)}& & \text{(Electric potential)}\\ \left(i\right)& \text{At the surface of A}& \left(i\right)& 0& \left(P\right)& \frac{\sigma }{{\in }_0}(\frac{a^2}{c}-\frac{b^2}{c}+c)\\ \left(ii\right)& \text{At the surface of B}& \left(ii\right)& \frac{\sigma }{{\in }_0}& \left(Q\right)& \frac{\sigma }{{\in }_0}(\frac{a^2}{b}-b+c)\\ \left(iii\right)& \text{At the surface of C}& \left(iii\right)& -\frac{\sigma }{{\in }_0}& \left(R\right)& \frac{\sigma }{{\in }_0}(a-b+c)\\ \left(iv\right)& \text{At the centre of spheres}& \left(iv\right)& \frac{2\sigma }{{\in }_0}& \left(S\right)& \frac{\sigma }{{\in }_0}(a-b-c)\end{array}$

Which of the following combination is correct for B ?

Sound signal is sent through a composite tube as shown in this fig. The radius of the semicircle is r. Speed of sound in air is v. The source of sound is capable to generate frequencies in the range $f_{1}to{f}_2(f_2>f_1)$. If n is an integer then, frequency for maximum intensity is given by

Two transverse waves in a given medium are represented by the curve as shown in the figure. Find the ratio of their intensities. All the measurements are in $\text{S.I}$ units.

Suppose you have three concave mirrors A, B, and C of focal lengths 10 cm, 15 cm and 20 cm. For each concave mirror, you perform the experiment of image formation for three values of an object distance of 10 cm, 20 cm and 30 cm. Giving reason and answer the following:

a) For the three object distance, identify the mirror/ mirrors which will form an image of magnification -1.

b) Out of the three mirrors identify the mirror which would be preferred to be used for shaving purposes/makeup.

A small quantity of a solution containing $N{a}^{24}$ radio – nuclide of half life T and activity $R_0$ is injected into blood of a person. $1c{m}^3$ of sample of blood taken from the blood of the person shows activity $R_1$. If the total volume of the blood in the body of the person is V, find the time after which sample is taken:

From a waterfall, water pours down at the rate of $100kg{s}^{-1}$ on the blades of a turbine. If the height of the fall is 120 m, what is the power delivered to the turbine?

The longest wavelength emitted in hydrogen atom belonging to Balmer series is ($R$ is Rydberg constant)

Why were electrons coming out from a metal when it was kept under a light source?

Two wires AC and BC are tied at C of small sphere of mass 5 kg, which revolves at a constant speed v in the horizontal circle of radius 1.6 m. Find the maximum value of v.

A person $AB$ of height $170\text{cm}$ is standing in front of a plane mirror. His eyes are at height $164\text{cm}$. At what distance from $P$ should a hole be made in the mirror so that he cannot see his hairs?

A $200\Omega$ resistor has a certain color code. If one replaces the red color by green in the code, the new resistance will be

A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $\rho =K{r}^a,$ where $K$ and $a$ are constants and $r$ is the radial distance from its centre. If the electric field at $r=\frac{R}{2}$ is $\frac{1}{8}$ times that at $r=R,$ the value of $a$ is (answer upto two decimal places)

A particle P is sliding down a frictionless hemispherical bowl. It passes the point A at t = 0. At this instant of time, the horizontal component of its velocity is v. A bead Q of the same mass as P is ejected from A at t = 0 along the horizontal string AB with speed v. Friction between the bead and the string may be neglected. Let t_P and t_Q be the respective times taken by P and Q to reach the point B. Then:

In the figure, all surfaces are smooth. Find the acceleration of the body and the force exerted by the floor on the body. (Take $g=10m/s^2$)

One mole of an ideal gas undergoes a process in which $T=T_0+a{V}^3$, where $T_0$ and '$a$' are positive constants and $V$ is the volume. The volume for which the pressure of the gas will be minimum is

Two blocks A and B of equal masses $m\text{kg}$ are suspended with the help of ideal pulleys and string arrangement as shown in figure. Then acceleration $\left({\text{in m/s}}^2\right)$ of mass B will be:

A black body at $227{}^{\circ }\text{C}$ radiates heat at the rate of $7{\text{cal/cm}}^2\text{s}$. At a temperature of $727{}^{\circ }\text{C}$, the rate of heat radiated in the same unit will be -