Direction (Read the following questions and choose) (A) If both Assertion and Reason are true and the Reason is correct explanation of assertion (B) If both Assertion and Reason are true, but reason is not correct explanation of the Assertion (C) If Assertion is true, but the Reason is false (D) If Assertion is false, but the Reason is true Assertion: The square of the period of revolution of a planet is proportional to the cube of its distance from the sun. Reason: Sun's gravitational field is inversely proportional to the square of its distance from the planet (a) \(\mathrm{A}\) (b) \(\mathrm{B}\) (c) \(\mathrm{C}\) (d) D

According to Kepler's Third Law of Planetary Motion, the square of the period of revolution of a planet (T²) is proportional to the cube of its average distance from the sun (r³). Mathematically, this can be expressed as: \[T^2 \propto r^3\] Given this law, the Assertion is true.

According to Newton's Law of Universal Gravitation, the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, the Sun's gravitational field (F) on a planet with mass m and distance r from the sun would be: \(F = G \frac{M_S m}{r^2}\), where \(M_S\) is the mass of the sun and G is the gravitational constant. So, the Sun's gravitational field is indeed inversely proportional to the square of its distance from the planet. Therefore, the Reason is also true.

Now we need to check if the Reason is the correct explanation of the Assertion. From both Kepler's Third Law and Newton's Law of Universal Gravitation, when we consider centripetal force to maintain the circular motion of a planet around the Sun: \( F = \frac{m v^2}{r} \) Newton's Law of Gravitation: \( F = G \frac{M_S m}{r^2} \) Combining these equations: \( \frac{m v^2}{r} = G \frac{M_S m}{r^2} \) The mass of the planet, m, and the distance from the Sun, r, both cancel out: \(v^2 = G \frac{M_S}{r}\) Since \(v = \frac{2\pi r}{T}\), we can substitute and obtain: \( \frac{4 \pi^2 r^2}{T^2} = G \frac{M_S}{r}\) Rearrange to get the relationship between the period T and the distance r: \(T^2 \propto r^3\) The gravitational relationship derived from Newton's Law does indeed correctly explain Kepler's Third Law as mentioned in the Assertion. Therefore, the Reason is the correct explanation for the Assertion.

Based on our analysis, we can conclude that: (A) Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion. So, the correct answer is (a) A.