Solution
This problem asks us to evaluate statements regarding a heavy rope (catenary) hanging between two points A and B.
(a) Horizontal component of tensile force is uniform.
Consider a free body diagram of any section of the rope. Since the only external force is gravity (vertical), there is no horizontal force to change the horizontal component of tension.
$$ T_x = \text{Constant} $$
Statement (a) is correct.
(b) Vertical component of tensile force increases with height.
Consider a section of rope from the lowest point to a height $y$. The vertical upward tension $T_y$ must support the weight of the rope segment below it. As we go higher, the length of the supported rope increases, so the weight increases.
$$ T_y = \lambda g s $$ (where $s$ is arc length from bottom).
Statement (b) is correct.
(c) Angle $\alpha$ can be greater than angle $\beta$.
It can not be true as the vertical(cos component) should be greater at A compared to B, as the weight of the rope has to be balanced.
(d) Angle $\alpha$ cannot assume a value of $0^\circ$.
For $\alpha$ to be $0^\circ$, the rope must be perfectly vertical at point A. This implies the horizontal component of tension $T_x = T \sin(0) = 0$. If horizontal tension is zero, the rope cannot span any horizontal distance. Since A and B are horizontally separated, $T_x$ must be non-zero.
Statement (d) is correct.