Introduction:
In this scenario, a glass hemisphere with a refractive index of 4/3 and a radius of 4 centimeters is placed on a plane mirror. A point object is placed at a distance 'd' on the axis of symmetry of the hemisphere. The objective is to determine the value of 'd' for which the final image formed by the glass hemisphere and the plane mirror is at infinity.

Understanding the Situation:
To solve this problem, we need to analyze the refraction and reflection of light at the interface between the glass hemisphere and the air, as well as the reflection of light at the plane mirror.

Refraction at the Glass Hemisphere:
When light enters the glass hemisphere from air, it bends towards the normal due to the higher refractive index of the glass. The refractive index of the glass hemisphere is given as 4/3.

Reflection at the Plane Mirror:
After passing through the glass hemisphere, the light reaches the plane mirror. At the plane mirror, the light undergoes reflection, following the law of reflection. The reflected light travels back through the glass hemisphere.

Formation of the Image:
The light rays that pass through the glass hemisphere and reflect off the plane mirror will form an image. To determine the location of the image, we can use the concept of virtual rays.

Virtual Rays:
Consider two virtual rays: one passing through the center of curvature of the hemisphere and the other passing through the point object. These rays will intersect at a point, which will be the position of the image formed by the glass hemisphere and the plane mirror.

Image at Infinity:
If the final image is at infinity, it means that the virtual rays are parallel to each other after passing through the glass hemisphere and reflecting off the plane mirror. This implies that the rays are incident at the critical angle on the interface between the glass and air.

Calculating the Critical Angle:
The critical angle can be calculated using Snell's law, which states that the sine of the angle of incidence divided by the sine of the angle of refraction is equal to the ratio of the refractive indices of the two media.

Conclusion:
To find the value of 'd' for which the final image formed by the glass hemisphere and the plane mirror is at infinity, we need to calculate the critical angle at the interface between the glass hemisphere and air. By determining the position of the point object relative to the glass hemisphere, we can calculate the distance 'd' required for the final image to be at infinity.