Correct Answer: $3\;\mathrm{cm}$

Solution : The area of a circle with $r$ as the original radius $= \pi r^2$
Given that the area of the circle is increased by $22\;\mathrm{cm^2}$ and the original radius is increased by $1\;\mathrm{cm}$.
The area of the circle after the radius is increased $=\pi (r+1)^2$
$⇒\pi (r+1)^2 = \pi r^2 + 22$
$⇒\pi r^2 + 2\pi r + \pi = \pi r^2 + 22$
$⇒2\pi r + \pi = 22$
$⇒(2 r + 1)\pi = 22$
$⇒(2 r + 1) = \frac{22}{\pi}$
$⇒2 r + 1  = 7$
$⇒r = 3$
Hence, the correct answer is $3\;\mathrm{cm}$.