In a school students at Pioneer career Kolkata wrote Mock test which has three subjects DI, VA and QA, here is the result of these students. 80 students cleared cut off in DI, 70 in VA and 60 in QA. Only 40 students cleared all the three subjects. 10 students failed to clear cut off even in one subjects. 50 students cleared cut off in VA and QA. 5 students cleared in cut off in only QA.
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.
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