A river flows from west to east at a speed of 5 metres per minute....

Let the width of the river be $d$.

The velocity of the man in still water is $v_{m} = 10$ m/min and the velocity of the river is $v_{r} = 5$ m/min.

Let the man swim at an angle $\theta$ with respect to the direction perpendicular to the river flow (due north).

The component of the man's velocity perpendicular to the river flow is $v_{m} \cos \theta$.

The time taken to cross the river is given by $t = \dfrac{d}{v_{m} \cos \theta}$.

To minimize the time $t$, the component $v_{m} \cos \theta$ must be maximum.

The maximum value of $\cos \theta$ is $1$, which occurs when $\theta = 0^{{}^{{}^{\circ}}}$.

Therefore, the man should swim in the direction perpendicular to the river flow, which is due north.