Q.If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law- (Note: more than one of the given options may be correct.)

(A) Elliptical orbits

Stable, closed orbits (ones that retrace the same path every revolution) exist only for very special force laws — a result known as Bertrand's theorem, which shows this happens only for the inverse-square law ($F\propto1/r^2$, giving Kepler's ellipses) and Hooke's law ($F\propto r$). For an inverse-cube law, the effective radial potential no longer has the right shape to produce a stable closed elliptical path — orbits instead spiral in or spiral out. So planets will not have elliptic orbits under an inverse-cube law. (A) is correct.

(B) Circular orbits

A circular orbit only requires that the gravitational force at radius $r$ supply exactly the centripetal force needed for that radius:

This equation has a valid solution for every $r>0$ — a circular orbit is mathematically possible at any radius. What changes under the inverse-cube law is stability: a small radial nudge doesn't oscillate back (as it does for inverse-square orbits) but instead grows, so the orbit is unstable, not impossible. Since option (B) claims circular orbits are not possible (rather than merely unstable), it is false.

(C) Projectile motion near Earth's surface …

Step 1: Bertrand's theorem -- only inverse-square/Hooke's-law forces give closed orbits for all energies; inverse-cube generally doesn't -- (a) true. Step 2: circular orbits ARE possible (though unstable) -- (b) false. Step 3: a hand-thrown stone covers negligible height vs Earth's radius, field looks locally uniform regardless of force law -- parabo …