Your questions are different and interesting.

(A) Despite what the children believe, shouting do not resolve the situation.
Wrong: Shouting does not.

(B) The gym teacher is ready to teach the class how to play archery, but displaying the proper techniques are always difficult for young children.
Wrong. displaying... IS...

(C) Because Mrs. Doe is a perfectionist, coloring eggs for the holidays is likely to take the entire afternoon.
Sounds better than all other answers. S-V agreement looks ok.

(D) Grading papers during the morning announcements are looked down upon.
Wrong. Grading papers is looked down upon.
(E) Telling the magicians' secrets to audiences are not allowed.
Wrong. Is not allowed.

Atul, You wrote chose the sentence

but both D and E are looking fine for me

Is grading a gerund ... should it have 'is' in D

IMO C looks better of all.
A-shouting does not
B-displaying is..
D-grading is
E-Telling is

Interesting...source?

Ah.. Nice replies from Rohu and vinesh.. Now I understood the verb-noun..
thanks guys

Only C looks fine .

(A) Despite what the children believe, shouting do not resolve the situation.

(B) The gym teacher is ready to teach the class how to play archery, but displaying the proper techniques are always difficult for young children.

(C) Because Mrs. Doe is a perfectionist, coloring eggs for the holidays is likely to take the entire afternoon.

(D) Grading papers during the morning announcements are looked down upon.

(E) Telling the magicians' secrets to audiences are not allowed

Hi guys the OA is C, but i thought that people will discuss the point of confusion which i had in my mind when i first saw this question...anyways let me show that point...

In all the choices except C there is a clear cut sub-verb agreement error...gerund has to take sing verb...but look at the choice C again, in this choice
"coloring eggs" for the holidays is likely to take the entire afternoon, in this choice i got confused initially because it seems to me like the subject is Eggs(plural) and coloring is an adjective so the subject has to be plural which is not there..because an -ing form can also work as adjective...but then after looking at the meaning i realize that its coloring that takes the entire afternoon not the eggs..so actually eggs is the adjective...

Thanks