A uniform magnetic field $$B$$ exist in the region between $$x=0$$ and $$x = {{3R} \over 2}$$ (region $$2$$ in the figure) pointing normally into the plane of the paper. A particle with charge $$+Q$$ and momentum $$p$$ directed along $$x$$-axis enters region $$2$$ from region $$1$$$ at point $${P_1}\left( y \right) = - R).$$ Which of the following option(s) is/are correct?

A rocket is launched normal to the surface of the Earth, away from the sun, along the line joining the Sun and the Earth. The Sun is $$3 \times 10{}^5$$ times heavier than the earth and is at a distance $$2.5 \times {10^4}$$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $${V_c} = 11.2km\,{s^{ - 1}}.$$. The minimum initial velocity $$\left( {{v_s}} \right)$$ required for the rocket to be able to leave the sun-earth system is closest to (Ignore the the rotation and revoluation of the earth and the presence of any other planet)

A symmetric star shaped conducting wire loop is carrying a steady state current $${\rm I}$$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $$4a.$$ The magnitude of the magnetic field at the center of the loop is

Consider regular polygons with number of sides $$n=3,4,5....$$ as shown in the figure. The center of mass of all the polygons is at height $$h$$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $$\Delta $$. Then $$\Delta $$ depends on $$n$$ and $$h$$ as