A glass sphere of refractive index `1.5` forms the real image of object `O` at point `I` as shown in the figure when kept in air of refractive index `1`. If the value of `x` is `kR`. Find `k`

Answer

5

Explanation

We can reverse the question. Let `I` behaves as object and `O` behaves as image, because retracing the path will not later the distances.
Consider first refraction at point `N`
`1.5/V-1.0/(-R)=(1.5-1.0)/(+R)implies1.5/v=1/(2R)-1/Rimpliesv=-3R`

Consider second refraction at point `M`, for this `I`, wil behaves normal object so
`1.0/v-1.5/(-5R)=(1.0-1.5)/(-R)`
`implies1/v+3/(10R)=1/(2R)=5/(10R)`
`implies 1/v=5/(10R)-3/(10R)=1/(5R)`
`implies v=5R`
`impliesx=5Rimpliesk=5`