$A$ star $2.5$ times the mass of the sun and collapsed to a size of $12 \;km$ rotates with a speed of $1.2 \;rev/s$. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (Mass of the sun $= 2 \times 10^{30} \;kg$)

Two particles,each of mass $m$,are moving in a circle of radius $R$ under the influence of their mutual gravitational attraction. What is their speed?

The orbital velocity of a body near the surface of a planet $A$ is equal to the escape velocity of a body from the planet $B$. If the masses of planets $A$ and $B$ are the same,the ratio of their radii is

Two bodies,each of mass $M$,are kept fixed with a separation $2L$. $A$ particle of mass $m$ is projected from the mid-point of the line joining their centres,perpendicular to the line. The gravitational constant is $G$. The correct statement$(s)$ is/are:
$(a)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $4 \sqrt{\frac{GM}{L}}$
$(b)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2 \sqrt{\frac{GM}{L}}$
$(c)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $\sqrt{\frac{2GM}{L}}$
$(d)$ The energy of the mass $m$ remains constant

$A$ spherical body of radius $R$ consists of a fluid of constant density $\rho$ and is in equilibrium under its own gravity. If $P(r)$ is the pressure at a distance $r$ from the center $(r < R)$,then the correct option$(s)$ is(are):
$(A) P(r=0) = P_c$ (maximum pressure at center)
$(B) \frac{P(r=3R/4)}{P(r=2R/3)} = \frac{63}{80}$
$(C) \frac{P(r=3R/5)}{P(r=2R/5)} = \frac{16}{21}$
$(D) \frac{P(r=R/2)}{P(r=R/3)} = \frac{20}{27}$

Every planet revolves around the sun in an elliptical orbit. Which of the following statements are correct?
$A.$ The force acting on a planet is inversely proportional to the square of the distance from the sun.
$B.$ The force acting on the planet is inversely proportional to the product of the masses of the planet and the sun.
$C.$ The centripetal force acting on the planet is directed away from the sun.
$D.$ The square of the time period of revolution of a planet around the sun is directly proportional to the cube of the semi-major axis of the elliptical orbit.
Choose the correct answer from the options given below: