What is the minimum number of students who did not clear cut off in exactly two subjects?

With the given information, we can draw the following Venn diagram:

Let the number of students who cleared the cut-off in exactly one, two and three subjects be a, b and c, respectively, and the number of students who cleared the cut-off in both VA and DI be y.

Thus, c = 40

a + 2b + 3c = 80 + 70 + 60 = 210

a + 2(5 + 10 + y) + 3(40) = 210

a + 2y = 210 - 120 - 30

a + 2y = 60

We have to find the minimum value of a.

'a' is at least 5 as the students in only QA = 5.

At a = 5, y = $$\frac{55}{2}$$. Since y needs to be an integer, a cannot be 5.

At a = 10, y = 25. But in this case, the total number of students in VA = 75. Thus, a cannot be 10.

At a = 20, y = 20. This case satisfies all the conditions.

Hence, the minimum number of students who did not clear cut off in exactly two subjects = Minimum number of students who cleared only one subject = 20.

The answer is option C.