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Question #41686

Difficulty: 8/10 | 🟢 Free

The Question

Consider the optimization problem:

Maximize $Z = 2[x] + 3y$

subject to the constraints: $2x + 3y \le 18$ $x \ge 0, y \ge 0$

where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $S$ be the set of all optimal solutions $(x, y)$ that maximize $Z$ . Which of the following statements is true?

Options

A.
$S$ contains exactly 2 points.
B.
$S$ contains exactly 10 points.
C.
$S$ is an infinite set of points lying on a line segment.
D.
$S$ contains exactly 4 points.

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Multiple Optimal Solutions|Intersection Point Accuracy|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question exploits a deeply ingrained conditioning from standard Linear Programming.

Students are taught that if an objective function is maximized at two vertices of a feasible region (here, $(0,6)$ and $(9,0)$ both yield $Z=18$ ), the entire line segment connecting them contains an infinite number of optimal solutions (Option C).

By introducing the Greatest Integer Function into the objective, the continuous plateau is shattered into discrete points where the fractional part of $x$ is zero.

A secondary trap (Option D) catches students who correctly identify the integer constraint on $x$ but falsely assume $y$ must also be an integer, which would limit the solutions to $x \in \{0, 3, 6, 9\}$ .

Quick Facts About This Question

Question Information

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