CAT 2025 Slot 3 QA Q14 Set Theory

Question 14 of the Quantitative Ability (QA) section, CAT 2025 Slot 3, with CAT's official answer and a worked solution.

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
  1. A
    30
  2. B
    35
  3. C
    40
  4. D
    55

Answer: (B) 35 Set Theory MCQ

Solution

  1. Let \(n(P \cap C) = n(C \cap M) = y\), \(n(P \cap M) = 2y\), all three \(= g \ge 1\).
  2. \(150 = 75 + 111 + 40 - 4y + g \Rightarrow g = 4y - 76\). \(g \ge 1\) forces \(y \ge 20\) (then \(g = 4\)).
  3. Physics but not maths \(= 75 - 2y\), largest at \(y = 20\): \(75 - 40 = 35\).
  4. Check regions at \(y = 20\), \(g = 4\): only P 19, only M 55, only C 4, P&M only 36, P&C only 16, C&M only 16 - all non-negative, total 150.

Same concept in other years

See every question on Three-set Venn diagrams: maximise or minimise a region.

More Set Theory questions from CAT

All 9 Set Theory questions asked since 2017.