The centre of mass of solid hemisphere of radius 8 cm is x from the centre of the flat surface. Then value of x is __________.

A Young's double-slit experiment is performed using monochromatic light of wavelength $$\lambda $$. The intensity of light at a point on the screen, where the path difference is $$\lambda $$, is K units. The intensity of light at a point where the path difference is $${\lambda \over 6}$$ is given by $${{nK} \over {12}}$$, where n is an integer. The value of n is __________.

Two planets have masses M and 16 M and their radii are $$a$$ and 2$$a$$, respectively. The separation between the centres of the planets is 10$$a$$. A body of mass m is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is :

Given the masses of various atomic particles
m_p = 1.0072 u, m_n = 1.0087 u, m_e = 0.000548 u,
$${m_{\overline v }}$$ = 0, m_d = 2.0141 u, where p $$ \equiv $$ proton, n $$ \equiv $$ neutron,
e $$ \equiv $$ electron, $$\overline v $$ $$ \equiv $$ antineutrino and d $$ \equiv $$ deuteron. Which of the following process is allowed by momentum and energy conservation?