Free particle wave function

yes! (actually I wrote it wrong.. ans is C if BC is not applied) But this is a Gate question and the Ans key is staying ans should be A.... We can only get to that ans if we apply ...

ok..... if we won't that b. c. then we use the defination of wf to get wf and and is c.

but, exp i(k.r-wt) is also a free. particle wave function.. then why we are discarding it? and we are not told here to Apply any B.C.. then why we are applying this B. C?

A ans is correct