What was the purpose of d-day?
a
to defend britain
b
to free jews from auschwitz
c
to overthrow mussolini
d
to invade france

The question is kind of tricky. D-Day was the code word for the Allied invasion of France, so D is definitely a possible answer. But the PURPOSE of that invasion was in part to defend Britain, which at the time was being threatened by Nazi air attacks and by a possible Nazi invasion. So A is possible as well.

The overthrow of Mussolini was certainly a factor in the Allied strategy, but the invasion of Italy, not France, was how that task was accomplished. So C is out.

There's competing evidence over how much the Allies knew about Auschwitz by the time of D-Day. Either way, the immediate purpose of D-Day was not to free the Jews from Auschwitz, though that was an ultimate result of D-Day. So B is out.

Since the question is asking the PURPOSE of D-Day, I think A is the best answer. D is more of a description of what D-Day WAS rather than an explanation of its PURPOSE.