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Question #41691

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{ (x,y) \in \mathbb{R}^2 : 3x + 4y \le 25, \; 4x + 3y \le 25, \; x + y \le 7, \; 0 \le x \le 6, \; 0 \le y \le 6 \}$ . Let $f: A \to B$ be a surjective function defined by $f(x,y) = 7x + 7y + 5$ . Which of the following intervals represents the set $B$ ?

Options

A.
$[5, 54]$
B.
$[5, 55]$
C.
$[5, 47]$
D.
$\left[5, \frac{148}{3}\right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Redundant Constraints|Surjective Mapping (Onto)|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

The trap exploits the student's tendency to assume that the intersection of the two 'most complex' constraints ( $3x+4y \le 25$ and $4x+3y \le 25$ ) automatically forms the optimal vertex of the feasible region.

Because $50/7 \approx 7.14$ is very close to 7, students might hastily conclude that $x+y \le 7$ is a redundant constraint and ignore it.

By doing so, they calculate the maximum at $(25/7, 25/7)$ , yielding an incorrect upper bound of 55.

This tests the rigorous verification of all constraints and the understanding that seemingly redundant constraints can actively truncate the feasible region and restrict the codomain of a surjective mapping.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Redundant Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free