Option C is the correct answer.

Instead of solving the problem as shown below, we can use logic to reach the solution. From theory, the value of acceleration due to gravity decreases as we move above or below the surface of the Earth. This means weight (W = mg) is maximum at the surface of the Earth. Hence, the weight of the given object must be less ($\rm < 800 N$) above the surface of the Earth. Upon looking at the options, we find that only 200 N satisfies the condition ($\rm 200 N < 800 N$). Therefore, it is the answer.

The value of acceleration due to gravity at a height h above the surface of the earth is given by the formula $\rm g \prime = g \left ( \frac{R^{2}}{ (R + h)^{2} }\right )$

Given

The radius of the Earth (R) = 6371 km

Height from the surface of the Earth (h) = 6371 km

(R + h) = (6371 + 6371) km = 12742 km

By using the formula,

$\rm g \prime = g \left ( \frac{6371^{2}}{ (12742)^{2}} \right )$

$\rm \therefore g \prime = g \cdot \frac{1}{4}$

The weight of an object at a given place is given by $\rm W = mg = 800 N$ (given), where m is the mass of the object (generally taken as constant) and g is the acceleration due to gravity at that place.

Let $\rm W \prime$ be the weight of the object at the height of $\rm 6371$ km from the surface of the Earth.

$\rm W \prime = m g \prime$

$\rm or, W \prime = m \cdot g \frac{1}{4}$

$\rm or, W \prime = (mg) \cdot \frac{1}{4}$

$\rm or, W \prime = 800 \cdot \frac{1}{4}$

$\rm \therefore W \prime = 200 N$

Hence, the required weight of the object at a height of 6371 km above the surface of the Earth is 200 N.