Answer : Option C

Explanation :

Let the distance be x km
if travels at 10 kmph, Arun will reach point A at 2 pm
if travels at 15 kmph, Arun will reach point 12 noon
=> Time taken when traveling at 10 km = Time taken when traveling at 15 km + 2 hours

$MF#%\begin{align}
&\Rightarrow \dfrac{x}{10} = \dfrac{x}{15} + 2\\\\
&\Rightarrow \dfrac{x}{10} - \dfrac{x}{15} = 2\\\\
&\Rightarrow 3x - 2x = 2 \times 30\\\\
&\Rightarrow x = 60\\\\
&\Rightarrow \text{Distance = 60 km}
&\end{align} $MF#%

$MF#%\begin{align}
&\Rightarrow \text{Time taken = }\dfrac{\text{Distance}}{\text{Speed}}= \dfrac{60}{10} = 6\text{ hours}\\\\
&\Rightarrow \text{He must have started 6 hours back 2 pm, ie, at 8 am}\\\\
&\Rightarrow \text{Now he wants to reach at 1 pm. ie; time to be taken = 5 hours}\\\\
&\Rightarrow \text{Speed needed = }\dfrac{\text{Distance}}{\text{Time}}= \dfrac{60}{5} = 12\text{ kmph}
\end{align} $MF#%