Physics Gravitation

The mass density of a spherical galaxy varies as K/r over a large distance 'r' from its center. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as:

A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is:
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The acceleration due to gravity on the earth's surface at the poles is g an and angular velocity of the earth about the axis passing through the pole is ω. An object is weight at the equator and at a height h above the poles by using a spring balance. If the weights are found to be same, then h is : (h << R, where R is the radius of the earth)
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Consider two solid spheres of radii ${}_{}$ $R_1=1\mathrm{m}$ , $R_2=2m$ ${}_{}$ and masses $M_1$ ${}_{}$ and $M_2$ ${}_{}$ , respectively. The gravitational field due to sphere (1) and (2) are shown. The value of $\frac{M_1}{M_2}$ $\frac{{}_{}}{{}_{}}$ is:

A satellite of mass $m$ $$ is launched vertically upwards with an initial speed $u$ $$ from the surface of the earth. After it reaches height $R(R=$ $$ radius of the earth), it ejects a rocket of mass $\frac{\mathrm{m}}{10}$ $\frac{}{}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ( $G$ $$ is the gravitational constant ; $$ $M$ is the mass of the earth)

A box weighs $196\mathrm{N}$ on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $\mathrm{g}=10{\mathrm{m}\mathrm{s}}^{-2}$ at the north pole and the radius of the earth $=6400\mathrm{k}\mathrm{m}$  )
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The maximum and minimum distances of a comet from the Sun are 1.6 x1012m and 8.0 x1010m respectively. If the speed of the comet at the nearest point is 6x104ms-1, the speed at the farthest point is:

Two identical particles of mass 1 kg each go round a circle of radius R, under the action of their mutual gravitational attraction. The angular speed of each particle is:
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The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of 9.0 × 10³ km. Find the mass of Mars.
{Given 4π²/G = 6 × 10¹¹ N⁻²kg²}