Gravitation · JEE Advanced previous year questions

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $r_2$ from the center of the earth, such that $r_1=1.21r_2$. The time period of the second satellite as measured from the geostationary satellite is $\frac{24}{p}\mathrm{hours}$. The value of $p$ is ____

Two satellites P and Q are moving in different circular orbits around the Earth (radius $R$). The heights of P and Q from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{P}}=R/3$. The accelerations of P and Q due to Earth's gravity are $g_{\mathrm{P}}$ and $g_{\mathrm{Q}}$, respectively. If $g_{\mathrm{P}}/g_{\mathrm{Q}}=36/25$, what is the value of $h_{\mathrm{Q}}$?

1. A.$3R/5$
2. B.$R/6$
3. C.$6R/5$
4. D.$5R/6$

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck's constant $h$ and the speed of light $c$, as $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$. Which of the following is the correct option?

1. A.$\alpha =7,\beta =-1,\gamma =-2$
2. B.$\alpha =-7,\beta =-1,\gamma =-2$
3. C.$\alpha =7,\beta =-1,\gamma =2$
4. D.$\alpha =-7,\beta =1,\gamma =-2$