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Question #40383

Difficulty: 8/10 | 🟢 Free

The Question

Let $U$ be a universal set of $60$ students. Let $A$ and $B$ be the sets of students who like Mathematics and Physics, respectively. It is given that $n(A \cap B) = 10$ . Let $x = n(A \cap B')$ and $y = n(A' \cap B)$ . The school administration imposes the constraints $2x + 3y \le 60$ and $x + y \ge 15$ . To optimize a resource metric, the function $Z = 4x + 6y$ is to be maximized. If $M$ is the maximum possible value of $Z$ , and $N$ is the number of distinct valid pairs $(x,y)$ that achieve this maximum, find the value of $M + N$ .

Options

A.
131
B.
122
C.
129
D.
120

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Venn Diagram Overlap Logic|Multiple Optimal Solutions|Non-Negativity Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the 'Multiple Optimal Solutions' trap in linear programming, disguised within a Venn diagram framework.

Students often evaluate the objective function only at the vertices of the feasible region, incorrectly assuming the maximum occurs at a single point (or just the two vertices), leading to $N=1$ or $N=2$ .

The Venn diagram context implicitly enforces the non-negativity and integer constraints ( $x, y \in \mathbb{Z}_{\ge 0}$ ), forcing students to find all lattice points along the boundary segment $2x + 3y = 60$ rather than treating it as a continuous real-valued line.

Failing to account for $k=0$ or $k=10$ (the boundary points) is another common non-negativity trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Venn Diagram Overlap Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free