A man wishes to cover 1 km distance in river water. In still water he takes 12 minutes to cover it, but in the flowing river he takes 13 minutes. The speed of the flowing water of the river is:

The correct option is 5/13 km/h.

Step 1: Calculate the speed of the man in still water
Given:
Distance covered, $d=1\text{km}$
Time taken in still water, $t_1=12\text{minutes}$

Convert the time into hours:
$t_1=\frac{12}{60}\text{hours}=\frac{1}{5}\text{hours}$

The speed of the man in still water ($v_m$) is:
$v_m=\frac{\text{Distance}}{\text{Time}}=\frac{1}{\frac{1}{5}}=5\text{km/h}$

Step 2: Calculate the net speed in flowing water
Since it takes longer to cover the same distance in the flowing river (13 minutes instead of 12 minutes), the man is swimming upstream (against the flow of the current).
Time taken in flowing river, $t_2=13\text{minutes}=\frac{13}{60}\text{hours}$

The effective upstream speed ($v_{\text{up}}$) is:
$v_{\text{up}}=\frac{\text{Distance}}{\text{Time}}=\frac{1}{\frac{13}{60}}=\frac{60}{13}\text{km/h}$

Step 3: Find the speed of the flowing water
The upstream speed is equal to the speed of the man in still water minus the speed of the river current ($v_r$):
$v_{\text{up}}=v_m-v_r$

Rearranging the formula to solve for $v_r$:
$v_r=v_m-v_{\text{up}}$

Substitute the calculated values into the formula:
$v_r=5-\frac{60}{13}$

Simplify the fraction:
$v_r=\frac{5\times 13-60}{13}=\frac{65-60}{13}=\frac{5}{13}\text{km/h}$

Thus, the speed of the flowing water of the river is 5/13 km/h.