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Question #41773

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(x,y)$ in the Cartesian plane satisfying the following inequalities: $\begin{vmatrix} 1 & 1 & 1 \\ x & 1 & 5 \\ y & 2 & 4 \end{vmatrix} \ge 0, \quad \begin{vmatrix} 1 & 1 & 1 \\ x & 3 & 5 \\ y & 8 & 4 \end{vmatrix} \ge 0, \quad \begin{vmatrix} 1 & 1 & 1 \\ x & 3 & 1 \\ y & 8 & 2 \end{vmatrix} \ge 0$ Find the minimum value of the objective function $Z = 2x + 3y$ on the region $S$ .

Options

A.
8
B.
20
C.
22
D.
30

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

Objective Function Gradient|Determinant as Geometric Area/Volume|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of oriented area and determinant properties in the context of linear programming.

The constraints are presented as determinants representing the signed area of triangles formed by a variable point and two fixed vertices.

A common trap is to extract the fixed points $(1,2), (5,4), (3,8)$ , assume they form a closed triangular feasible region, and simply evaluate the objective function at these vertices (yielding the incorrect minimum of 8).

However, the second determinant has its fixed points swapped compared to a standard counter-clockwise orientation, which flips the sign of the determinant (Row/Column Exchange Parity).

This reverses the inequality, making the feasible region an unbounded area outside the triangle.

Students must rigorously expand the determinants and verify the region, rewarding algebraic precision over geometric assumptions.

Quick Facts About This Question

Question Information

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