Set Theory CAT PYQs , 18 Questions with Solutions

In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is

In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

In a small college, students are allowed to take only one specialization. Traditionally, only two specializations are offered: Science and Arts. Students enrolled to specialize in Science must take Physics and Mathematics subjects, while students enrolled to specialize in Arts must take Economics and Political Science subjects. Students enrolled in Science are not allowed to take either Economics or Political Science, while students enrolled in Arts are not allowed to take either Physics or Mathematics. Recently, the college has started a third specialization called MatEco that requires students to take Economics and Mathematics. However, MatEco students would not be allowed to take either Physics or Political Science. When the college opens this new specialization for enrolment, it allows students, originally enrolled in Science or Arts, to switch to MatEco. From among the students originally enrolled in Arts, 20 students switch to MatEco. This makes the number of Science students twice the number of Arts students. After this, from among the students who originally enrolled in Science, 45 students switch to MatEco. This makes the number of Arts students twice the number of Science students. In total, how many students, from among those originally enrolled in Science or Arts, are now taking Economics?

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

In a college consisting of 200 students, 112 students took the Maths Olympiad, 160 students took the Physics Olympiad and 128 students took the Chemistry Olympiad. If each student in the college takes at least one of the three exams, what is the minimum number of students who took all the three exams?

Among 100 students, x₁ have birthdays in January, x₂ have birthdays in February, and so on. If x₀ = max(x₁, x₂, ..., x₁₂), then the smallest possible value of x₀ is

Students in a college have to choose at least two subjects from chemistry, mathematics and physics. The number of students choosing all three subjects is 18, choosing mathematics as one of their subjects is 23 and choosing physics as one of their subjects is 25. The smallest possible number of students who could choose chemistry as one of their subjects is

A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, 58 members can play both football and tennis, 25 can play both cricket and tennis, while 63 can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is

A bag contains marbles of three colours-red, blue and green. There are $8$ blue marbles in the bag.

There are two additional statement of facts available:

$1)$ If we pull out marbles from the bag at random, to guarantee that we have at least $3$ green marbles, we need to extract $17$ marbles.

$2)$ If we pull out marbles from the bag at random, to guarantee that we have at least $2$ red marbles, we need to extract $19$ marbles.

Which of the two statements above, alone or in combination shall be sufficient to answer the question "how many green marbles are there in the bag"?

In 2015, 100 aspirants appeared for an exam. They had to answer four sections: Maths, DI, LR and English. Qualified in Maths = 55. Qualified in LR = 38. Qualified in (Maths+English) = 30. Qualified in (LR+English) = 15. Qualified in (Maths+LR) = 20. Qualified in (Maths+LR+English) = 5. Qualified in DI = 22. Qualified in (DI+LR) = 5. Qualified in (DI+Maths) = 5. Qualified in (DI+Maths+LR) = 5. Qualified in English = 50. Those who qualified in English could not qualify in DI.

Each of 74 students in a class studies at least one of the three subjects H, E and P. Ten students study all three subjects, while twenty study H and E, but not P. Every student who studies P also studies H or E or both. If the number of students studying H equals that studying E, then the number of students studying H is?

If among 200 students, 105 like pizza and 134 like burger, then the number of students who like only burger can possibly be

If A = {6²n - 35n - 1 : n = 1, 2, 3, ...} and B = {35(n - 1) : n = 1, 2, 3, ...} then which of the following is true?

For two sets A and B, let A △ B denote the set of elements which belong to A or B but not both. If P = {1,2,3,4}, Q = {2,3,5,6,}, R = {1,3,7,8,9}, S = {2,4,9,10}, then the number of elements in (P △ Q) △ (R △ S) is

In the final semester, an engineering college offers three elective courses and one mandatory course. A student has to register for exactly three courses: two electives and the mandatory course. The registration in three of the four courses is: $45$, $55$ and $70$. What will be the number of students in the elective with the lowest registration?

If A is the set of odd integers less than 50 and B is the set of the squares of integers in A, how many elements does the intersection of A and B contain?

In an international school, 80% of the students know French, 85% know German and 75% know both French and German. If 40 students don't know both, then find total number of students.

In a class of 60, along with English as a common subject, students can opt to major in Mathematics, Physics, Biology or a combination of any two. 6 students major in both Mathematics and Physics, 15 major in both Physics and Biology, but no one majors in both Mathematics and Biology. In an English test, the average mark scored by students majoring in Mathematics is 45 and that of students majoring in Biology is 60. However, the combined average mark in English, of students of these two majors, is 50. What is the maximum possible number of students who major ONLY in Physics?