Step 1: Meaning of mole fraction.
Mole fraction of the solute simply asks what share of all the moles in the mixture belong to the solute. To get it we need the moles of solute and the moles of water.
Step 2: Read the molality.
A 2.0 molal solution means 2.0 mol of solute are dissolved in exactly 1 kg, that is 1000 g, of water. So the solute part is already known, 2.0 mol.
Step 3: Convert the water mass to moles.
Water has a molar mass of 18 g/mol, so the moles of water are
\[ \frac{1000}{18} = 55.56\ \text{mol} \]
Step 4: Put it into the fraction.
\[ x_{solute} = \frac{2.0}{2.0 + 55.56} = \frac{2.0}{57.56} = 0.0347 \]
\[ \boxed{\text{Option (C): } x \approx 0.0347} \]