Since we are given, the sum of first 11 terms of an arithmetic progression is 275. We can take the first term to be 'a' and the subsequent terms with a common difference of 'd', added to the previous term in the sequence.
Hence, the terms would be {a, a+d, a+2d, ..., a+10d} when the series has 11 terms in it.
Similarly, when the series would be having 12 terms, the relevant terms would be {a, a+d, ..., a+11d}
Utilizing the formula of AP, we know sum of 'n' terms would be S = n/2 * [2a + (n-1) * d]
Substituting n = 11, we get the following.
S = 11/2 [2a + 10d] = 275
Simplifying this equation, we get the value of a+5d to be 25.
Now, since we are being asked to determine the sum of 12 terms in the series, we can represent it as the following.
(Sum of 11 terms in the series + 12th term)
So, this would be (275 + a + 11d). Since we know, the value of a + 5d is 25. We can represent it as, 300 + 6d.
Hence, we know the sum of 12 terms in the series would be more than 300 and also the common difference would be a multiple of 6 (it is also given that the common difference is a positive integer).
Reviewing the answer choices given, the answer would be 'D'.