Given below are two statements:
Statement-I: An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable.
Statement-II: Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight when the elevator goes down with increasing speed.
In the light of the above statements, choose the correct answer from the options given below:

Let the mass of the elevator be $$M$$ and the magnitude of its velocity be $$v$$.

For an entire elevator moving vertically, the only forces acting are:
  • its weight $$W = Mg$$ acting downward,
  • the tension $$T$$ in the supporting cable acting upward.

By Newton’s second law the net force along the vertical is
$$T - Mg = M a$$ $$-(1)$$
where $$a$$ is the elevator’s vertical acceleration (taken positive upward).

Case 1: Elevator moving with uniform speed (either upward or downward).
Uniform speed means $$a = 0$$. Substituting $$a = 0$$ in $$(1)$$ gives
$$T - Mg = 0 \; \Longrightarrow \; T = Mg$$ $$-(2)$$

Equation $$(2)$$ shows that whenever the tension equals the weight, the elevator’s acceleration is zero, so it can remain at rest or travel upward/downward with constant speed. Hence Statement I is true.

Case 2: Person of mass $$m$$ standing on the elevator floor while the elevator goes down with increasing speed.
“Down with increasing speed” means the elevator has an acceleration of magnitude $$a$$ directed downward, i.e. upward acceleration is $$-a$$.

For the person, the forces are:
  • true weight $$mg$$ downward,
  • normal reaction $$N$$ from the floor upward.

Taking upward as positive, Newton’s second law for the person gives
$$N - mg = m(-a)$$ $$-(3)$$

Simplifying $$(3)$$,
$$N = m(g - a)$$ $$-(4)$$

Since the elevator is speeding up downward, $$a \gt 0$$, so from $$(4)$$:
$$g - a \lt g \; \Longrightarrow \; N \lt mg$$

The normal force (apparent weight) is less than the actual weight when the elevator accelerates downward. Therefore Statement II is false.

Combining the results: Statement I is true and Statement II is false. According to the given options, this corresponds to Option B.

Answer: Option B