The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity

The correct option is: "cannot be zero at any point."

Step-by-Step Explanation:

1. Nature of Gravitational Force:
Gravity is a purely attractive force. According to Newton's law of gravitation, every mass element in the Earth exerts an attractive pull on any object placed on the surface. Because negative mass does not exist, there are no repulsive gravitational forces.

2. Geometry of a Surface Point:
For any point located on the surface of the approximate sphere representing the Earth, the entire mass of the Earth lies on one side of the tangent plane at that point. In other words, all mass elements of the Earth lie in the hemisphere beneath the surface observer.

3. Vector Sum of Acceleration:
The net acceleration due to gravity $\mathbf{g}$ at a surface point is the integral (vector sum) of the gravitational contributions from all mass elements $dm$ within the Earth:
$\mathbf{g}=\int \frac{Gdm}{r^2}\hat{\mathbf{r}}$
where $G$ is the gravitational constant, $r$ is the distance to the mass element, and $\hat{\mathbf{r}}$ is the unit vector pointing towards the mass element.

4. Impossibility of Zero Gravity on the Surface:
Since all mass elements lie entirely on one side of the tangent plane at the surface point, every single force vector points towards the interior of the Earth. The components of these force vectors normal to the tangent plane all point in the same direction (inward). Because there is no mass outside the Earth's surface to exert an outward gravitational pull, these inward components cannot be cancelled out. Consequently, the net acceleration due to gravity on the surface must have a non-zero magnitude pointing inward, meaning it cannot be zero at any point on the surface.