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Question #41670

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the feasible region in the $xy$ -plane defined by $x \ge 0, y \ge 0$ and the determinant constraints: $\begin{vmatrix} [x] & [y] \\ 1 & 2 \end{vmatrix} \le 202, \quad \begin{vmatrix} [x] & -[y] \\ 2 & 1 \end{vmatrix} \le 202$ where $[ \cdot ]$ denotes the greatest integer function. If $M$ is the supremum of the objective function $Z = 4x + 3y$ for $(x,y) \in R$ , find the value of $M$ .

Options

A.
604
B.
606
C.
611
D.
613

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

Feasible Region Constancy|Determinant as Geometric Area/Volume|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle piecewise constant optimization within a continuous Linear Programming Problem (LPP).

The inclusion of the Greatest Integer Function inside the determinant constraints forces the feasible region to be a union of discrete unit squares, turning the boundary into a 'staircase'.

A common trap is ignoring the GIF and solving the continuous LPP to get 606, or finding the maximum integer coordinates (121, 40) and incorrectly concluding the maximum is 604.

Students must recognize that while $[x]$ and $[y]$ are bounded by integers, the continuous variables $x$ and $y$ can approach $[x]+1$ and $[y]+1$ , requiring them to add the supremum of the fractional parts to the discrete maximum.

Quick Facts About This Question

Question Information

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