Point $Q$ will touch again, when cone will roll around $P$ and will travel arc length of $2\pi r$
So, arc length made by cone $= 2 \pi r = 10\pi $ cm
If the cone rotates one round around point $P$ it will cover perimeter of length $2 \pi l$ cm
where, $l = \sqrt{h^2+r^2} = 13$ cm
So, perimeter of one rotation $= 26 \pi$ cm
Thus, the angle(in radian) which cone makes $= \frac{10\pi}{26\pi}\times 360\times \frac{\pi}{180} = \frac{10\pi}{13}$
Correct Answer: $D$