Class JEE Mathematics Sets, Relations, and Functions Q #1137
COMPETENCY BASED
APPLY
4 Marks 2024 JEE Main 2024 (Online) 4th April Morning Shift NUMERICAL
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.
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Step-by-Step Solution

  1. Let M, P, and C denote the sets of students studying Mathematics, Physics, and Chemistry, respectively. Let $|M|$, $|P|$, and $|C|$ denote the number of students in each set.
  2. Given information:
    • $125 \le |M| \le 130$
    • $85 \le |P| \le 95$
    • $75 \le |C| \le 90$
    • $|P \cap C| = 30$
    • $|C \cap M| = 50$
    • $|M \cap P| = 40$
    • Total students = 220
    • Students studying none = 10
    • So, $|M \cup P \cup C| = 220 - 10 = 210$
  3. We know that $|M \cup P \cup C| = |M| + |P| + |C| - |M \cap P| - |P \cap C| - |C \cap M| + |M \cap P \cap C|$
  4. $210 = |M| + |P| + |C| - 40 - 30 - 50 + |M \cap P \cap C|$
  5. $210 = |M| + |P| + |C| - 120 + |M \cap P \cap C|$
  6. $|M \cap P \cap C| = 330 - (|M| + |P| + |C|)$
  7. To find the least value of $|M \cap P \cap C|$ (m), we need to maximize $|M| + |P| + |C|$.
    • $|M|_{max} = 130$, $|P|_{max} = 95$, $|C|_{max} = 90$
    • $|M| + |P| + |C|_{max} = 130 + 95 + 90 = 315$
    • $m = 330 - 315 = 15$
  8. To find the most value of $|M \cap P \cap C|$ (n), we need to minimize $|M| + |P| + |C|$.
    • $|M|_{min} = 125$, $|P|_{min} = 85$, $|C|_{min} = 75$
    • $|M| + |P| + |C|_{min} = 125 + 85 + 75 = 285$
    • $n = 330 - 285 = 45$
  9. Therefore, $m + n = 15 + 45 = 60$

Correct Answer: 60

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply their understanding of set theory and Venn diagrams to solve a real-world problem involving overlapping sets of students studying different subjects.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of the concepts of sets, subsets, union, intersection, and Venn diagrams to determine the minimum and maximum number of students studying all three subjects.
Syllabus Audit: In the context of JEE, this is classified as COMPETENCY. It assesses the ability to apply set theory principles to solve a problem, rather than just recalling definitions or formulas.

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