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Question #41692

Difficulty: 8/10 | 🟢 Free

The Question

Let $c$ be the number of positive integers $t$ satisfying $\frac{(t-1)(t-5)}{t-3} \le 0$ , and $d$ be the number of positive integers $t$ satisfying $\frac{t^2 - 7t + 12}{t - 2} \le 0$ . Let $S$ be the region in the $xy$ -plane defined by the inequalities $x \ge 0$ , $y \ge 0$ , $[x] + [y] \le 4$ , and $2[x] - [y] \ge 1$ , where $[ \cdot ]$ denotes the greatest integer function. If $M$ is the supremum of the objective function $Z = cx + dy$ over the region $S$ , find the value of $M$ .

Options

A.
12
B.
15
C.
18
D.
24

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

Corner Point Method Precision|Objective Function Gradient|Wavy Curve (Method of Intervals)|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question fuses the Wavy Curve method with a discontinuous Linear Programming problem.

The primary trap tests the misinterpretation of Greatest Integer Function (GIF) boundaries.

If a student drops the GIF and solves the continuous LP system ( $x+y \le 4$ and $2x-y \ge 1$ ), the vertices yield a maximum $Z$ of 12.

Another trap is assuming $[x]+[y] \le 4$ implies $x+y < 5$ , yielding 15.

The correct approach requires evaluating the discrete integer blocks $(X,Y)$ and recognizing that $x$ and $y$ can independently approach their upper bounds $X+1$ and $Y+1$ , allowing $x+y$ to approach 6.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Corner Point Method Precision
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free