Let P\left( r \right) = {Q \over {\pi {R^4}}}r be the change density distribution for a solid sphere of radius $$R$$ and total charge $$Q$$. For a point $${}^{\prime }{p}^{\prime }$$ inside the sphere at distance $$r_1$$ from the center of the sphere, the magnitude of electric field is :

Two points $$P$$ and $$Q$$ are maintained at the potentials of $$10$$ $$V$$ and $$-4$$ $$V$$, respectively. The work done in moving $$100$$ electrons from $$P$$ to $$Q$$ is :

(This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.) Statement-1 : For a charged particle moving from point $$P$$ to point $$Q$$, the net work done by an electrostatic field on the particle is independent of the path connecting point $$P$$ to point $$Q.$$ Statement-2 : The net work done by a conservative force on an object moving along a closed loop is zero.

A particle has an initial velocity $$3\hat{i}+4\hat{j}$$ and an acceleration of $$0.4\hat{i}+0.3\hat{j}$$. Its speed after 10 s is:

Consider a rubber ball freely falling from a height $$h=4.9$$ $$m$$ onto a horizontal elastic plate. Assume that the duration of collision is negligible and the collision with the plate is totally elastic. Then the velocity as a function of time and the height as a function of time will be :

The surface of a metal is illuminated with the light of $$400$$ $$nm.$$ The kinetic energy of the ejected photoelectrons was found to be $$1.68$$ $$eV.$$ The work function of the metal is : $$\left(hc=1240eV.nm\right)$$

The height at which the acceleration due to gravity becomes {g \over 9} (where $$g=$$ the acceleration due to gravity on the surface of the earth) in terms of $$R,$$ the radius of the earth, is:

A mixture of light, consisting of wavelength $$590$$ $$nm$$ and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the screen. The central maximum of both lights coincide. Further, it is observed that the third bright fringe of known light coincides with the $$4$$th bright fringe of the unknown light. From this data, the wavelength of the unknown light is :

Three sound waves of equal amplitudes have frequencies $$\left(v-1\right),v,\left(v+1\right).$$ They superpose to give beats. The number of beats produced per second will be :

A motor cycle starts from rest and accelerates along a straight path at $$2m/s^2.$$ At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at $$94\%$$ of its value when the motor cycle was at rest? (Speed of sound $$=330m{s}^{-1}$$)

A current loop $$ABCD$$ is held fixed on the plane of the paper as shown in the figure. The arcs $$BC$$ (radius $$=b$$) and $$DA$$ (radius $$=a$$) of the loop are joined by two straight wires $$AB$$ and $$CD$$. A steady current $$I$$ is flowing in the loop. Angle made by $$AB$$ and $$CD$$ at the origin $$O$$ is $$30^{\circ }.$$ Another straight thin wire steady current $$I_1$$ flowing out of the plane of the paper is kept at the origin. The magnitude of the magnetic field $$(B)$$ due to the loop $$ABCD$$ at the origin $$(O)$$ is :

A current loop $$ABCD$$ is held fixed on the plane of the paper as shown in the figure. The arcs $$BC$$ (radius $$=b$$) and $$DA$$ (radius $$=a$$) of the loop are joined by two straight wires $$AB$$ and $$CD$$. A steady current $$I$$ is flowing in the loop. Angle made by $$AB$$ and $$CD$$ at the origin $$O$$ is $$30^{\circ }.$$ Another straight thin wire steady current $$I_1$$ flowing out of the plane of the paper is kept at the origin. Due to the presence of the current $$I_1$$ at the origin:

A thin uniform rod of length $$l$$ and mass $$m$$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $$\omega$$. Its center of mass rises to a maximum height of:

Which of the following pairs represents linkage isomers ?

Which of the following has an optical isomer ?

Solid Ba(NO3)2 is gradually dissolved in a 1.0 $$\times$$ 10-4 M Na2CO3 solution. At what concentration of Ba2+ will a precipitate begin to form ? (Ksp for BaCO3 = 5.1 $$\times$$ 10−9 )

Calculate the wavelength (in nanometer) associated with a proton moving at 1.0 x 103 ms−1 (Mass of proton = 1.67 $$\times$$ 10-27 kg and h = 6.63 $$\times$$ 10-34 Js) :

In an atom, an electron is moving with a speed of 600 m/s with an accuracy of 0.005%. Certainity with which the position of the electron can be located is (h = 6.6 $$\times$$ 10-34 kg m2s-1, mass of electron, em = 9.1 $$\times$$ 10-31 kg)

In which of the following arrangements, the sequence is not strictly according to the property written against it

The set representing the correct order of ionic radius is :