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Question #41768

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{F}(x) = \int (2x \hat{i} + 2x \hat{j}) dx$ . The constant of integration is $\vec{K} = C \hat{i} - C \hat{j}$ , where $C$ is an integer. Let $\vec{V}_1 = \vec{F}(1)$ and $\vec{V}_2 = \vec{F}(2)$ . Consider the Linear Programming Problem: Maximize $Z = x + y$ Subject to: $\vec{r} \cdot \vec{V}_1 \ge 12$ $\vec{r} \cdot \vec{V}_2 \le 18$ $x \ge 0, y \ge 0$ where $\vec{r} = x \hat{i} + y \hat{j}$ . Let $S$ be the set of all integers $C \in [-10, 10]$ for which the feasible region is non-empty. Find the number of elements in $S$ .

Options

A.
12
B.
21
C.
10
D.
14

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

Unbounded Region Trap|Linear Independence vs Dependence|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Integration Constant Trap' combined with an 'Unbounded/Infeasible Region Trap'.

Students typically solve the boundary equations, find the objective function value $Z = x+y = 2$ , and incorrectly assume the feasible region is non-empty for all $C$ , leading to the trap answer 21.

However, the integration constant $C$ shifts the slopes and intercepts of the constraints.

By forcing the intersection point to be a function of $C$ , the feasible region completely collapses and becomes empty when $C$ crosses the critical thresholds (here, $|C| < 5$ ).

This tests the student's rigorous geometric understanding of LPP constraints beyond mere algebraic substitution.

Quick Facts About This Question

Question Information

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