An object is placed infront of a convex mirror at a distance of  50cm. A plane mirror is  introduced covering the lower half of the convex mirror. If the distance between the  object and plane mirror is 30cm, it is found that there is no parallax between the images  formed by the two mirrors. The radius of curvature of the convex mirror is _____cm.

The plane mirror will form erect and virtual image of same size at a distance of 30cm  behind it. So the distance of image formed by the plane mirror from the convex mirror  will be 
 $PI=MI-MP=MO-MP=30-20=10cm$
Now as this image coincide with image formed by the convex mirror for convex mirror 
$$\begin{gathered} u=-50cm,v=10cm \\ \frac{1}{f}=\frac{1}{10}+\frac{1}{-50}\Rightarrow f=12.5cm \\ R=2f=2\times 12.5=25cm \end{gathered}$$