13. When does a protostar become a star?
a. when it begins to fuse hydrogen in its core
b. when its mass becomes greater than half a solar mass
c. when the peak of its spectrum moves into visible light
d. when the cloud of gas and dust around it is cleared away

Hydrogen fusion in its core is what defines a Main Sequence star. The lower mass limit for stars is much lower than half a solar mass (it's about 1/10 of a solar mass). A star can have the peak of its blackbody spectrum in the infrared and still be a star, and the cloud of gas and dust clears away after the star begins to fuse.

14. Which phase in a low-mass star's evolution lasts longest?
a. protostar
b. Main Sequence
c. red giant
d. horizontal branch

This is an important fact about the lives of stars that you should know. All of the other phases of evolution mentioned here are much shorter than the Main Sequence phase.

15. What is the difference between a red giant and an asymptotic giant?
a. an asymptotic giant's interior is like a red giant's, but its surface is hotter, so it's not red
b. high-mass stars become red giants; low-mass stars become asymptotic giants
c. an asymptotic giant is a star that is traveling up the giant branch and will eventually become a red giant
d. a red giant has a degenerate helium core; an asymptotic giant has a degenerate carbon core

The red giant phase in a low-mass star's evolution is reached after the supply of hydrogen in the core is exhausted. Then the star starts burning helium into carbon as a horizontal branch star. After the helium supply is exhausted, the star moves up the giant branch again, taking a slightly different path. This is called the asymptotic giant branch.

16. The Pleiades and the Hyades are two star clusters in the constellation of Taurus. The Pleiades has many bright blue stars but no bright red stars, and the Hyades has many bright red stars and few bright blue stars. What can we conclude?
a. the Pleiades is older than the Hyades
b. the Hyades is older than the Pleiades
c. the Pleiades is closer than the Hyades
d. the Hyades is closer than the Pleiades

Bright blue stars are Main Sequence stars. For a cluster to have them, it must be young enough that those stars have not left the Main Sequence yet. Bright red stars are red giants or asymptotic giants. For a cluster to have those, it must be old enough that some stars have left the Main Sequence. Thus, the Hyades is older than the Pleiades.