Half-lives of two radioactive elements $$A$$ and $$B$$ are $$20$$ minutes and $$40$$ minutes, respectively. Initially, the samples have equal number of nuclei. After $$80$$ minutes, the ratio of decayed number of $$A$$ and $$B$$ nuclei will be:

A pendulum clock loses $$12$$ $$s$$ a day if the temperature is $$40^{\circ }C$$ and gains $$4$$ $$s$$ a day if the temperature is $$20^{\circ }C.$$ The temperature at which the clock will show correct time, and the co-efficient of linear expansion $$\left(\alpha \right)$$ of the metal of the pendulum shaft are respectively :

An ideal gas undergoes a quasi static, reversible process in which its molar heat capacity $$C$$ remains constant. If during this process the relation of pressure $$P$$ and volume $$V$$ is given by $$P{V}^n=$$ constant, then $$n$$ is given by (Here $$C_p$$ and $$C_v$$ are molar specific heat at constant pressure and constant volume, respectively:

Which of the following shows the correct relationship between the pressure ‘P’ and density $$\rho$$ of an ideal gas at constant temperature ?

A Carnot freezer takes heat from water at 0oC inside it and rejects it to the room at a temperature of 27oC. The latent heat of ice is 336×103 J kg−1. If 5 kg of water at 0oC is converted into ice at 0oC by the freezer, then the energy consumed by the freezer is close to :

A simple pendulum made of a bob of mass m and a metallic wire of negligible mass has time period 2 s at T=0oC. If the temperature of the wire is increased and the corresponding change in its time period is plotted against its temperature, the resulting graph is a line of slope S. If the coefficient of linear expansion of metal is $$\alpha$$ then the value of S is :

The ratio of work done by an ideal monoatomic gas to the heat supplied to it in an isobaric process is :

200 g water is heated from 40oC to 60oC. Ignoring the slight expansion of water, the change in its internal energy is close to (Given specific heat of water = 4184 J/kg/K) :

$${}^{\prime }{n}^{\prime }$$ moles of an ideal gas undergoes a process $$A$$ $$\to$$ $$B$$ as shown in the figure. The maximum temperature of the gas during the process will be :

The region between two concentric spheres of radii $${}^{\prime }{a}^{\prime }$$ and $${}^{\prime }{b}^{\prime },$$ respectively (see figure), have volume charge density \rho = {A \over r}, where $$A$$ is a constant and $$r$$ is the distance from the center. A such that the electric field in the region between the spheres will be constant, is :

Within a spherical charge distribution of charge density $$\rho$$(r), N equipotential surfaces of potential V0, V0 + $$\Delta$$V, V0 + 2$$\Delta$$V, .......... V0 + N$$\Delta$$V ($$\Delta$$ V > 0), are drawn and have increasing radii r0, r1, r2,..........rN, respectively. If the difference in the radii of the surfaces is constant for all values of V0 and $$\Delta$$V then :

The potential (in volts) of a charge distribution is given by. V(z) = 30 $$-$$ 5x2 for $$\left|z\right|$$ $$\le$$ 1 m. V(z) = 35 $$-$$ 10 $$\left|z\right|$$ for $$\left|z\right|$$ $$\ge$$1 m. V(z) does not depend on x and y. If this potential is generated by a constant charge per unit volume $${\rho }_0$$ (in units of $${\epsilon }_0$$) which is spread over a certain region, then choose the correct statement.

Radiation of wavelength $$\lambda ,$$ is incident on a photocell. The fastest emitted electron has speed $$v.$$ If the wavelength is changed to {{3\lambda } \over 4}, the speed of the fastest emitted electron will be:

A photoelectric surface is illuminated successively by monochromatic light of wavelengths $$\lambda$$ and {\lambda \over 2}. If the maximum kinetic energy of the emitted photoelectrons in the second case is 3 times that in the first case, the work function of the surface is :

When photons of wavelength $${\lambda }_1$$ are incident on an isolated sphere, the corresponding stopping potential is found to be V. When photons of wavelength $${\lambda }_2$$ are used, the corresponding stopping potential was thrice that of the above value. If light of wavelength $${\lambda }_3$$ is used then find the stopping potential for this case :

Hysteresis loops for two magnetic materials $$A$$ and $$B$$ are given below : These materials are used to make magnets for electric generators, transformer core and electromagnet core. Then it is proper to use :

A fighter plane of length 20 m, wing span (distance from tip of one wing to the tip of the other wing) of 15 m and height 5 m is flying towards east over Delhi. Its speed is 240 ms−1. The earth’s magnetic field over Delhi is 5 $$\times$$10−5 T with the declination angle ~ 0o and dip of $$\theta$$ such that sin $$\theta$$ = 2/3 . If the voltage developed is VB between the lower and upper side of the plane and VW between the tips of the wings then VB and VW are close to :

A satellite is revolving in a circular orbit at a height $${}^{\prime }{h}^{\prime }$$ from the earth's surface (radius of earth $$R;h<<R$$). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field, is close to : (Neglect the effect of atmosphere.)

An astronaut of mass m is working on a satellite orbiting the earth at a distance h from the earth’s surface. The radius of the earth is R, while its mass is M. The gravitational pull FG on the astronaut is :

Figure shows elliptical path abcd of a planet around the sun S such that the area of triangle csa is {1 \over 4} the area of the ellipse. (See figure) With db as the semimajor axis, and ca as the semiminor axis. If t1 is the time taken for planet to go over path abc and t2 for path taken over cda then :