Among the Group 13 elements, which of the following burns in ${\mathrm{N}}_2$ at high temperature, forming compound $\mathrm{MN}$? 

Most group 13 elements do not react with $N_2$ under normal conditions since $N_2$ molecule has a strong triple bond which is difficult to break for most group 13 elements.

Correct answer: Option (2)

Aluminium reacts with $N_2$ through the following chemical reaction:

$6\mathrm{Al}+3{\mathrm{ }\mathrm{N}}_2\to 6\mathrm{AlN}$

Nitrogen is very less reactive with most of the elements. For reaction to happen, atomic nitrogen is required, but under normal conditions, atomic nitrogen doesn't exist. Instead, it exists in the form of $N_2,$ which has a triple bond. This triple bond makes it very difficult for $N_2$ to react.

But aluminium has higher affinity for $N_2$, still, this reaction takes place at around $800-1200°\mathrm{C}$. This high temperature is required since a layer of oxygen covers the surface of aluminium by forming $A{l}_2{O}_3$. Hence, option (2) is correct.

Incorrect answer:

Option (1): As discussed above, the reactivity of $N_2$ is extremely low because of the triple bond present there. Moreover, Indium doesn't have much affinity towards nitrogen and oxygen. Hence, this reaction doesn't take place. Hence, this option is incorrect.

Option (3): Same is the case with Gallium as well, it is not having enough affinity towards nitrogen. Hence, the reaction of Gallium with $N_2$ is generally carried out under drastic conditions of temperature and pressure. These reactions can never take place under normal reaction conditions. Hence, generally it may be considered that this element does not give reaction with $N_2$. This option is incorrect.

Option (4): Thallium also does not react with the $N_2$ gas. Hence, we can say that only Aluminum can form the compound AlN with $N_2$ only. Hence, this option is incorrect.

Among the given elements, Aluminium burns in $N_2$ to form $AlN$. Hence, option (2) is the correct answer.