Consider a transparent homogenous hemisphere of refractive index `n=2` in front of which a small object is placed in air as shown in figure.

Options

A. if value of `x=3R` then the final image of the small object OA will be real & inverted.
B. If value of `x=2R` then the final image of the small object OA will be inverted and magnified.
C. consider a ray starting from O which strikes the spherical surface at grazing incidence `(i~=90^(@))` taking `x=R` the angle (from normal) at which the ray will emerge from the plane surface is `30^(@)`
D. consider a ray starting from O which strikes the spherical surface at grazing incidence `(i~=90^(@))`. Takin `x=R` the angle (from normal) at which the ray will emerge from the plane surface is `90^(@)`.

Answer

A, D

Explanation

For first surface
`(2)/(V)-(1)/(-3R)=(2-1)/(R)implies(2)/(V)=(1)/(R)-(1)/(3R)`
`(2)/(V)=(2)/(3R)impliesV=3R`
for second surface `u=2R`
`(1)/(V)=(2)/(2R)impliesV=R` Real & inverted.
(B) for first surface
`(2)/(V)-(1)/(-2R)=(2-1)/(R)implies(2)/(V)=(1)/(R)-(1)/(2R)`
`impliesV=4Rimpliesm_(1)=-1`
for second surface `u=3R`
`(1)/(V)=(2)/(3R)impliesV=(3R)/(2)` real & inverted and same size.