A real rope is hanging by one end from the ceiling. The other end dangles freely. If the mass of the rope is 100 g, then the tension is(A) 0.98 N along the entire length of the rope. (B) 0.49 N along the entire length of the rope.(C) 0.98 N at the bottom of the rope, and varies linearly to zero at the top of the rope.(D) 0.98 N at the top of the rope, and varies linearly to zero at the bottom of the rope

The correct option is:

One should consider the effect of weight of the rope at various positions to account for the tension locally.
At the top, the magnitude of tension would be equal to the magnitude of weight because the entire mass of the rope lies below it, and can be given as:

Therefore the magnitude of tension at the top is 0.98 N.
As you traverse down the rope, the effective mass of the rope which lies in the rope below is reduced; therefore the magnitude of the tension keeps on decreasing. One must understand that the tension in the rope appear because the gravity acts on the rope, and the magnitude of this action depends on the mass of the rope below the point (on a rope) in consideration.
At the bottom, there is no weight below to create tension in the string, and therefore this point is free from any tension.
Thus, (d) is the correct option while the others are ruled out.