$A$ solid sphere of mass $M$ and radius $a$ is surrounded by a uniform concentric spherical shell of thickness $2a$ and mass $2M$. The gravitational field at distance $3a$ from the centre will be

$A$ spherical shell is divided into two parts. If the gravitational intensity at point $P$ due to the upper part is ${I_1}$ and due to the lower part is ${I_2}$,then:

Gravitational potential in a region is given by $V = -(x + y + z) \text{ J/kg}$. Find the gravitational intensity at $(2, 2, 2)$ in $\text{N/kg}$.

$A$ metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is:

This question contains Statement-$1$ and Statement-$2$. Of the four choices given after the statements,choose the one that best describes the two statements.
Statement-$1$: For a mass $M$ kept at the centre of a cube of side '$a$',the flux of the gravitational field passing through its sides is $-4\pi GM$.
Statement-$2$: If the direction of a field due to a point source is radial and its dependence on the distance '$r$' from the source is given as $\frac{1}{r^2}$,its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.

The net gravitational force at the centre of a square is found to be $F_{1}$ when four particles having masses $M, 2M, 3M,$ and $4M$ are placed at the four corners of the square as shown in the figure. It is $F_{2}$ when the positions of $3M$ and $4M$ are interchanged. The ratio $\frac{F_{1}}{F_{2}}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is . . . . . . .