For beams breadth is constant, A. Depth $${\text{d}} \propto {\text{M}}$$ B. Depth $${\text{d}} \propto \sqrt {\text{M}} $$ C. Depth $${\text{d}} \propto 3\sqrt {\text{M}} $$ D. Depth $${\text{d}} \propto \frac{1}{{\text{M}}}$$

The correct answer is: A. Depth $${\text{d}} \propto {\text{M}}$$

The depth of a beam is proportional to its bending moment, which is the force that causes the beam to bend. The greater the bending moment, the deeper the beam must be to resist the bending.

The breadth of a beam is the width of the beam. The breadth of a beam does not affect the depth of the beam, as long as the material of the beam is strong enough to support the bending moment.

The options B, C, and D are incorrect because they do not take into account the fact that the breadth of a beam is constant.