B

Step 1: Determine the relative speed components. The man's swimming speed in still water is 4 km/h, and the river's current speed is 3 km/h. Since he swims directly across (normal to the current), we need to use right triangle components to find the effective speed across the width of the river.

Step 2: Calculate the time taken to swim across the river. The distance across the river (width) is 1 km. The speed across the river is still 4 km/h since he swims perpendicular to the current. The time taken to cross the river is given by:
$$ ext{Time} = rac{ ext{Distance}}{ ext{Speed}} = rac{1 ext{ km}}{4 ext{ km/h}} = 0.25 ext{ hours} ext{ or } 15 ext{ minutes}. $$

Step 3: Calculate how far down the river he goes. In 0.25 hours, the river flows downstream at 3 km/h. Therefore, the distance flowed downstream is:
$$ ext{Distance downriver} = ext{Speed} imes ext{Time} = 3 ext{ km/h} imes 0.25 ext{ hours} = 0.75 ext{ km}. $$

Final Answer: The man takes 15 minutes to cross the river and drifts 0.75 km downstream.