If two planets are at mean distances \(d_1\) and \(d_2\) from the sun and their frequencies are \(n_1\) and \(n_2\) respectively, then:
1. \(n^2_1d^2_1= n_2d^2_2\)
2. \(n^2_2d^3_2= n^2_1d^3_1\)
3. \(n_1d^2_1= n_2d^2_2\)
4. \(n^2_1d_1= n^2_2d_2\)

A planet is revolving around the sun as shown in an elliptical path.

The correct option is:

(1)   The time taken in travelling DAB is less than that for BCD

(2)   The time taken in travelling DAB is greater than that for BCD

(3)   The time taken in travelling CDA is less than that for ABC

(4)   The time taken in travelling CDA is greater than that for ABC

In an elliptical orbit under gravitational force, in general

(1) Tangential velocity is constant

(2) Angular velocity is constant

(3) Radial velocity is constant

(4) Areal velocity is constant

Earth is revolving around the sun, if the distance of the Earth from the Sun is reduced to 1/4^th of the present distance then the present year length reduced to -

(1)$\frac{1}{4}$                                        

(2)$\frac{1}{2}$  

(3)$\frac{1}{8}$                                        

(4)$\frac{1}{6}$  

Two satellite are revolving around the earth with velocities $v_1$ and $v_2$ and in radii $r_1$ and $r_2\left(r_1>r_2\right)$respectively. Then -

(1)$v_1=v_2$                                

(2)$v_1>v_2$  

(3)$v_1<v_2$                                

(4)$\frac{v_1}{r_1}=\frac{v_2}{r_2}$  

If orbital velocity of planet is given by $v=G^a{M}^b{R}^c$, then -

(1)$a=1/3, b=1/3, c=‐1/3$      

(2)$a=1/2, b=1/2, c=‐1/2$      

(3)$a=1/2, b=‐1/2, c=1/2$   

(4)$a=1/2, b=‐1/2, c=‐1/2$          

The mass of a planet that has a moon whose time period and orbital radius are T and R respectively can be written as -

(1)$4{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$                                

(2)$8{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$  

(3)$12{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$                               

(4)$16{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$  

The eccentricity of earth's orbit is 0.0167. The ratio of its maximum speed in its orbit to its minimum speed is

(a)   2.507                                             (b)   1.033

(c)   8.324                                             (d)   1.000

Assuming the earth to have a constant density, point out which of the following curves show the variation of acceleration due to gravity from the centre of earth to the points far away from the surface of earth -