An object is placed in front of a convex mirror at a distance of 50cm plane mirror is introduced in front of the convex mirror converting the lower half of it. If the distance between the object and the plane mirror is 30cm and it is found that there is no parallax between the images formed by the two mirrors for the same object. What is the radius of curvature of the convex mirror?

Let O be the object placed in front of a convex mirror M1 at a distance of 50cm as shown in figure 5.105. The distance of the plane mirror M2 from the object is 30cm. We know that in a plane mirror the image is formed behind the mirror at the same distance as the object in front of it. It is also given that there is no parallax between the images formed by the two mirror, thus the image produced by the convex mirror is formed at a distance of 10cm behind the convex mirror which will coincide with the image produced by plane mirror at a distance 30cm behind at position Q as shown in figure below.
    
For reflection at convex mirror using coordinate sign convenition for mirror formula we have
u = 50cm, v = -10cm
using mirror formula, we have
$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$
$\frac{1}{f}=\frac{1}{50}-\frac{1}{10}=-\frac{4}{50}$
$f=-\frac{50}{4}$
so radius of curvature of the mirror is $R=2f=-\frac{50\times 2}{4}=-25cm$ 

the radius of curvature of convex mirror is 25cm.