In all the problems, the wires carry current.
Problem A (2 wires going into the page, find net mag. field at point P )
Problem B (A loop of wire, find net mag. field at point C)
Problem C (2 Wires with current opposite directions, find mag. field at point C)
My impression is that Bio-Savart is only used for finding the magnetic field of a wire at a point, where as the equation $B=\left(\frac{\mu_0}{2\pi}\right)\frac{I}{r}$ is used to find the magnetic field of a wire on another wire.
So in problem A, you can think of point P as another wire going into the page or out (doesn't matter). So you use $B=\left(\frac{\mu_0}{2\pi}\right)\frac{I}{r}$.
In problem B, you are finding the net magnetic field of 2 wires (the other 2 sides of the loop can be ignored) on a point. So you use Biot-Savart. With Biot-Savart you get $B_{net}$=$\frac{\mu_0I\theta}{4\pi}\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$. With $B=\left(\frac{\mu_0}{2\pi}\right)\frac{I}{r}$, you get a different result.
But the real difficulty is in deciding which to use for problem C, the question asks for the net magnetic field of 2 wires at a field between them. In this case, unlike in problem A, you can't think of the point as a wire (in or out of the page). My professor uses $B=\left(\frac{\mu_0}{2\pi}\right)\frac{I}{r}$.
Doesn't the assumption that the wires are of infinite length affect the calculation?