Two non-conducting spheres of radii $${R_1}$$ and $${R_2}$$ and carrying uniform volume charge densities $$ + \rho $$ and $$ - \rho ,$$ respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region

Two bodies, each of mass M, are kept fixed with a separation $$2L$$. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

A particle of mass m is attached to one end of a mass-less spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t = 0 with an initial velocity u₀. When the speed of the particle is 0.5 u₀. It collides elastically with a rigid wall. After this collision,

The magnitude of the normal reaction that acts on the block at the point Q is

The speed of the block when it reaches the point Q is

Match List I with List II and select the correct answer using the codes given below the lists:

List I

P. Boltzmann Constant
Q. Coefficient of viscosity
R. Plank Constant
S. Thermal conductivity

List II

1. [ML²T^-1]
2. [ML^-1T^-1]
3. [MLT^-3K^-1]
4. [ML²T^-2K^-1]

Using the expression $$2d\sin \theta = \lambda $$, one calculates the values of d by measuring the corresponding angles $$\theta $$ in the range 0 to 90^o. The wavelength $$\lambda $$ is exactly known and the error in $$\theta $$ is constant for all values of $$\theta $$. As $$\theta $$ increases from 0^o

A steady current I flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is placed coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and carries a steady current I. Consider a point P at a distance r from the common axis. The correct statement(s) is(are)

Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f₁. An observer in the other vehicle hears the frequency of the whistle to be f₂. The speed of sound in still air is V. The correct statement(s) is(are)

The figure below shows the variation of specific heat capacity (C) of a solid as a function of temperature (T). The temperature is increased continuously from 0 to 500 K at a constant rate. Ignoring any volume change, the following statement(s) is(are) correct to a reasonable approximation.

The radius of the orbit of an electron in a hydrogen-like atom is 4.5a₀, where a₀ is the Bohr radius. Its orbital angular momentum is $${{3h} \over {2\pi }}$$. It is given that h is Planck constant and R is Rydberg constant. The possible wavelength(s), when the atom de-excites, is(are)

If the direct transmission method with a cable of resistance 0.4 $$\Omega$$ km^$$-$$1 is used, the power dissipation (in %) during transmission is

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is 1 : 10. If the power of the consumers has to be supplied at 200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is

The correct statement is

The kinetic energy (in keV) of the alpha particle, when the nucleus $$_{84}^{210}Po$$ at rest undergoes alpha decay, is

A right-angled prism of refractive index $$\mu$$₁ is placed in a rectangular block of refractive index $$\mu$$₂, which is surrounded by a medium of refractive index $$\mu$$₃, as shown in the figure. A ray of light e enters the rectangular block at normal incidence. Depending upon the relationships between $$\mu$$₁, $$\mu$$₂ and $$\mu$$₃, it takes one of the four possible paths 'ef', 'eg', 'eh' or 'ei'.

Match the paths in List I with conditions of refractive indices in List II and select the correct answer using the codes given below the lists:

	List I		List II
P.	$$e \to f$$	1.	$${\mu _1} > \sqrt 2 {\mu _2}$$
Q.	$$e \to g$$	2.	$${\mu _2} > {\mu _1}$$ and $${\mu _2} > {\mu _3}$$
R.	$$e \to h$$	3.	$${\mu _1} = {\mu _2}$$
S.	$$e \to i$$	4.	$${\mu _2} < {\mu _1} < \sqrt 2 {\mu _2}$$ and $${\mu _2} > {\mu _3}$$

One mole of a monatomic ideal gas is taken along two cyclic processes E $$\to$$ F $$\to$$ G $$\to$$ E and E $$\to$$ F $$\to$$ H $$\to$$ E as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic.

Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using the codes given below the lists :

	List I		List II
P.	$$G \to E$$	1.	160$${P_0}{V_0}$$ln2
Q.	$$G \to H$$	2.	36$${P_0}{V_0}$$
R.	$$F \to H$$	3.	24$${P_0}{V_0}$$
S.	$$F \to G$$	4.	31$${P_0}{V_0}$$

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :

	List I		List II
P.	Alpha decay	1.	$$_8^{15}O \to _7^{15}N + ...$$
Q.	$${\beta ^ + }$$ decay	2.	$$_{91}^{238}U \to _{90}^{234}Th + ...$$
R.	Fission	3.	$$_{83}^{185}Bi \to _{82}^{184}Pb + ...$$
S.	Proton emission	4.	$$_{94}^{239}Pu \to _{57}^{140}La + ...$$