Both a spherical mirror and a thin spherical lens have a focal length of $-15 \; cm$. What type of mirror and lens are they?

What is the mirror formula for a spherical mirror?

$(a)$ Water has a refractive index of $1.33$ and alcohol has a refractive index of $1.36$. Which of the two media is optically denser? Give a reason for your answer. Draw a ray diagram to show the path of a ray of light passing obliquely from water to alcohol.
$(b)$ The absolute refractive index of diamond is $2.42$ and the absolute refractive index of glass is $1.50$. Find the refractive index of diamond with respect to glass.

Draw ray diagrams showing the image formation by a convex mirror when an object is placed:
$(a)$ at infinity
$(b)$ at a finite distance from the mirror

$(i)$ Define the principal focus of a concave mirror.
$(ii)$ Why do we prefer a convex mirror as a rear-view mirror in vehicles?
$(iii)$ We wish to obtain an erect image of an object using a convex lens. Draw a ray diagram to show the image formation in this case.

For what position of the object does a concave mirror form a real and diminished image?