A

Step 1: Determine the speed of the man and the river.
The speed of the river ($v_r$) is 5 m/min (eastwards) and the speed of the man ($v_m$) is 10 m/min (in still water).

Step 2: To swim directly across the river (north to south), the man has to counteract the eastward current of the river.
Using vector decomposition, the man must swim at an angle $ heta$ such that his effective velocity across the river would negate the river's current.

Step 3: The man's velocity can be decomposed into two components:
- One component ($v_{m_y}$) directed across the river (north-south) and the other component ($v_{m_x}$) directed against the river current (west-east).
These components can be expressed as follows:
- $v_{m_x} = v_m imes ext{sin}( heta)$
- $v_{m_y} = v_m imes ext{cos}( heta)$

Step 4: To counter the river's current, $v_{m_x}$ must equal the river's speed:
$$ v_m imes ext{sin}( heta) = v_r$$
which gives us:
$$ ext{sin}( heta) = \frac{v_r}{v_m} = \frac{5}{10} = 0.5$$
This means $ heta = 30^{ ext{o}}$ with respect to the north direction (the direction across the river).

Step 5: The direction the man should swim is northwest, which means swimming at an angle of 30 degrees on the western side (to compensate for the river's eastward flow).

Therefore, the direction he should swim is to the northwest.