Let the speed of the boat in still water $=x$ km/hr

Speed of the current = 2 km/hr

Then, speed downstream $=\left(x+2\right)$ km/hr

speed upstream $=\left(x-2\right)$ km/hr

Total time taken to travel 10 km upstream and back = 55 minutes $=\dfrac{55}{60}$ hour = $\dfrac{11}{12}$ hour

$\Rightarrow \dfrac{10}{x-2} + \dfrac{10}{x+2} = \dfrac{11}{12}$

$120\left(x+2\right) + 120\left(x-2\right) = 11\left(x^2-4\right)$

$240x = 11x^2 - 44$

$11x^2 - 240x - 44 = 0$

$11x^2 - 242x +2x - 44 = 0$

$11x\left(x-22\right)+2\left(x-22\right)=0 $

$ \left(x-22\right)(11x+2)=0$

$x=22\text{ or }\dfrac{-2}{11}$

Since $x$ cannot be negative, $x$ = 22

i.e., speed of the boat in still water = 22 km/hr