A piece of wire is bent in the shape of a parabola y = kx² (y-axis vertical) with a bead of mass m on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with a constant acceleration $$a$$. The distance of the new equilibrium position of the bead, where the bead can stays at rest with respect to the wire, from the y-axis is

Photoelectric effect experiments are performed using three different metal plates p, q and r having work functions $$\phi_p=2.0~\mathrm{eV}$$, $$\phi_q=2.5~\mathrm{eV}$$ and $$\phi_r=3.0~\mathrm{eV}$$, respecticely. A light beam containing wavelengths of 550 nm, 450 nm and 350 nm with equal intensities illuminates each of the plates. The correct I-V graph for the experiment is (Take hc = 1240 eV nm)

The mass M shown in the figure below oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $$k$$. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $$\theta$$ in one direction and released. The frequency of oscillation is

The figure shows the PV plot of an ideal gas taken through a cycle ABCDA. The part ABC is a semicircle and CDA is half of an ellipse. Then,

Under the influence of the Coulomb field of charge +Q, a charge $$-$$q is moving around it in an elliptical orbit. Find out the correct statement(s):

Two metallic rings A and B, identical in shape and size but having different resistivities $$\rho_A$$ and $$\rho_B$$, are kept on top of two identical solenoids as shown in the figure below. When current I is switched on in both the solenoids in identical manner, the rings A and B jump to heights $$h_A$$ and $$h_B$$, respectively, with $$h_A > h_B$$. The possible relation(s) between their resistivities and their masses $$m_A$$ and $$m_B$$ is (are)

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure below, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then,

A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air-column is the second resonance. Then,

Column II gives certain systems undergoing a process. Column I suggests changes in some of the parameters related to the system. Match the statements in Column I to the appropriate process(es) from Column II:

	Column I		Column II
(A)	The energy of the system is increased.	(P)	System : A capacitor, initially uncharged. Process : It is connected to a battery.
(B)	Mechanical energy is provided to the system, which is converted into energy of random motion of its parts.	(Q)	System : A gas in an adiabatic container filled with an adiabatic piston. Process : The gas is compressed by pushing the piston.
(C)	Internal energy of the system is converted into its mechanical energy.	(R)	System : A gas in a rigid container. Process : The gas gets cooled due to colder atmosphere surrounding it.
(D)	Mass of the system is decreased.	(S)	System : A heavy nucleus, initially at rest. Process : The nucleus fissions into two fragments of nearly equal masses and some neutrons are emitted.
		(T)	System : A resistive wire loop. Process : The loop is placed in a time varying magnetic field perpendicular to its plane.

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits S$$_1$$ and S$$_2$$. In each of these cases, S$$_1$$P$$_0$$ = S$$_2$$P$$_0$$, S$$_1$$P$$_1$$ $$-$$ S$$_2$$P$$_1$$ = $$\lambda/4$$ and S$$_1$$P$$_2$$ $$-$$ S$$_2$$P$$_2$$ = $$\lambda/3$$, where $$\lambda$$ is the wavelength of the light used. In the cases B, C and D, a transparent sheet of refractive index $$\mu$$ and thickness t is pasted on slit S$$_2$$. The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point P on the screen from the two slits is denoted by $$\delta$$(P) and the intensity by I(P). Match each situation given in Column I with the statement(s) in Column II valid for that situation:

	Column I		Column II
(A)		(P)	$$\delta ({P_0}) = 0$$
(B)	$$(\mu-1)t=\lambda/4$$	(Q)	$$\delta ({P_1}) = 0$$
(C)	$$(\mu-1)t=\lambda/2$$	(R)	$$I({P_1}) = 0$$
(D)	$$(\mu-1)t=3\lambda/4$$	(S)	$$I({P_0}) > I({P_1})$$
		(T)	$$I({P_2}) > I({P_1})$$

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s², find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest.

Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m$$^2$$. The radii of bubbles A and B are 2 cm and 4 cm, respectively. Surface tension of the soap-water used to make bubbles is 0.04 N/m. Find the ratio $$n_B/n_A$$, where $$n_A$$ and $$n_B$$ are the number of moles of air in bubbles A and B, respectively. (Neglect the effect of gravity.)

A steady current I goes through a wire loop PQR having shape of a right angle triangle wit6h PQ = 3, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is $$k\left( {{{{\mu _0}I} \over {48\pi x}}} \right)$$, find the value of $$k$$.

A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening of the orifice. (Take atmospheric pressure = 1.0 $$\times$$ 10$$^5$$ N/m$$^2$$, density of water = 1000 kg/m$$^3$$ and g = 10 m/s$$^2$$. Neglect any effect of surface tension.)

A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. find the separation (in cm) between the successive nodes on the string.

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 distance from its centre. If the electric field at $$r = R/2$$ is 1/8 times than at $$r = R$$, find the value of $$a$$.

A metal rod AB of length 10x has its one end A in ice at 0$$^\circ$$C and the other end B in water at 100$$^\circ$$C. If a point P on the rod is maintained at 400$$^\circ$$C, then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is 540 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of $$\lambda x$$ from the ice end A, find the value of $$\lambda$$. (Neglect any heat loss to the surrounding.)

Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C.