Every free question on Venn Diagrams & Set Theory (Reasoning) from past WBCS Preliminary Examination and WBCS Main Examination papers, newest first, each with its answer and a short explanation. Free to read.
Answer: (A) 10%
Candidates who passed in at least one subject = 60 + 70 − 40 = 90%. So those who failed in both = 100 − 90 = 10%. 40% and 30% are the shares who failed in English and in Maths taken one at a time, not those who failed in both.
Answer: (A) 265
In the diagram the Hindi circle contains 70 (Hindi only) + 65 (Hindi and Bengali) + 53 (all three) + 47 (Hindi and English) = 235 persons. Those who do not know Hindi are 500 − 235 = 265, which is also the sum of the remaining regions 120 + 62 + 83. 200, 255 and 201 miss one or more of these regions.
Answer: (A) 17
Doctors who are neither players nor artists lie in the part of the triangle outside both the circle and the rectangle, which holds 17. The 8 are doctors who are artists, and 4 and 3 are doctors who are players.
Answer: (C) 53
Persons who know all three languages are in the central region common to the three circles, which shows 53. 65, 62 and 47 are the regions common to only two languages (Hindi–Bengali, Bengali–English and Hindi–English).
Answer: (C) 25
Artists who are players are in the overlap of the rectangle and the circle: 22 (not doctors) plus 3 (also doctors) = 25. 29 and 30 wrongly add the 4 or 8 regions, which lie outside that overlap.
Answer: (A) 5000
83% passed in at least one subject, so those who passed both = 58 + 68 − 83 = 43%. If 43% is 2150 candidates, the total is 2150 ÷ 0·43 = 5000. 6000 would make 43% equal 2580.
Answer: (B) 24
People who had at least one type = 60 + 30 − 15 = 75%, so 25% had neither. 25% of 96 = 24. Forgetting to subtract the 15% overlap would wrongly give 10%.
Answer: (D)
"Books" is the widest class, so it is the large circle. Every Bengali book is a book and every novel is a book, so both small circles must lie wholly inside the large one. Some Bengali books are novels (and some novels are Bengali books), so the two small circles must overlap each other. Only the last diagram shows this. The first diagram would suit two unrelated sub-classes such as "books, novels, dictionaries"; the diagrams in which a small circle crosses the boundary of the large one would wrongly mean that some Bengali books or novels are not books.
Answer: (D) $\dfrac{5}{27}$
From the diagram, 100 people know all three languages. Those who do not know Sanskrit at all are those in the Hindi-only, English-only, and Hindi-and-English-only regions: $200+220+120 = 540$. So the required ratio is $\dfrac{100}{540} = \dfrac{5}{27}$.
Answer: (A) b
The required region must lie inside the circle for Indians and the circle for historians but outside the circle for politicians. In the figure of the question paper that region is marked 'b'. The region common to all three circles would represent Indian historians who are also politicians, and so is excluded.
