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Question #41682

Difficulty: 8/10 | 🟢 Free

The Question

Let the feasible region $R$ in the $xy$ -plane be defined by the system of inequalities: $x \ge 0, \quad y \ge 0$ $x - y \le 2$ $x + \sqrt{3}y \le 6$ Let $S$ be the set of all values $\theta \in [0, 2\pi)$ such that the objective function $Z = x \cos \theta + y \sin \theta$ attains its maximum value on $R$ at infinitely many points. Find the value of $\sum_{\theta \in S} \theta$ .

Options

A.
$\frac{55\pi}{12}$
B.
$\frac{43\pi}{12}$
C.
$\frac{31\pi}{12}$
D.
$\frac{19\pi}{12}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Multiple Optimal Solutions|Objective Function Gradient|Normal Form Parameters

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of multiple optimal solutions in Linear Programming.

The core principle is that the objective function's gradient must align with the outward normal of a binding constraint for a maximum to occur along an edge.

The primary trap lies in incorrectly identifying the outward normals, particularly for the non-negativity constraints ( $x \ge 0, y \ge 0$ ), where students often mistakenly use the inward normals $(1,0)$ and $(0,1)$ yielding $\theta=0, \pi/2$ .

Another trap is miscalculating the normal form angle for $x - y \le 2$ by taking the wrong quadrant (e.g., $3\pi/4$ instead of $7\pi/4$ ).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Multiple Optimal Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free