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Question #41675

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = t\hat{i} + \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + t\hat{j} + \hat{k}$ , and $\vec{c} = \hat{i} + \hat{j} + t\hat{k}$ be three linearly dependent vectors, where $t$ is a real number such that $t < 0$ . Consider the following Linear Programming Problem:

Maximize $Z = 4x + 3y$ Subject to the constraints: $\vec{r} \cdot (\hat{i} + \hat{j}) \ge 1$ $\vec{r} \cdot (t\hat{i} + \hat{j}) \le 4$ $x \ge 0, y \ge 0$ where $\vec{r} = x\hat{i} + y\hat{j}$ .

The maximum value of $Z$ is:

Options

A.
12
B.
16
C.
4
D.
Does not exist

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

Unbounded Region Trap|Linear Independence vs Dependence|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate vector linear dependence with Linear Programming, specifically targeting the 'Unbounded Region Trap'.

By embedding a parameter $t$ within the constraints, students must first solve a determinant equation to find $t = -2$ .

The negative sign of $t$ creates a constraint ( $-2x + y \le 4$ ) that opens the feasible region to infinity along the positive x-axis.

The classic trap is that students will mechanically calculate the intersection points $(1,0), (0,1), (0,4)$ , evaluate the objective function, and incorrectly select the maximum corner-point value ( $12$ ).

This exposes the flaw of blindly applying the corner-point theorem without verifying the boundedness of the feasible region, a critical discriminator in JEE Advanced.

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