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Question #40318

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(x,y) \in \mathbb{Z}^2$ that simultaneously satisfy the following four inequalities: I. $\max(y - x, 4 - y) \ge 0$ II. $\max(y + x, 4 - y) \ge 0$ III. $\max(y - x, y - 5) \ge 0$ IV. $\max(y + x, y - 5) \ge 0$

Evaluate the sum: $\sum_{(x,y) \in S} 2^{-y}$

Options

A.
6
B.
$\frac{83}{16}$
C.
$\frac{13}{16}$
D.
$\infty$

Check Your Answer

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

Set Identity Laws|Redundant Constraints|Unbounded Region Trap|Sigma Notation Properties

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply Set Identity Laws (specifically the Distributive Law) to simplify a system of logical inequalities.

The primary trap is the 'Unbounded Region Trap' combined with 'Redundant Constraints'.

Students often mistakenly assume that the presence of the terms $4-y$ and $y-5$ implies the region is bounded by $y \le 4$ .

If they fail to simplify the logical propositions $(A \lor C) \land (A \lor \neg C) \equiv A$ , they will incorrectly intersect the region with $y \le 4$ , leading to a finite sum over a bounded triangle (yielding the distractor 83/16).

Recognizing the redundancy reveals the region is unbounded ( $y \ge |x|$ ), requiring the evaluation of a convergent infinite double geometric series (or an arithmetico-geometric progression), testing their rigor with Sigma notation properties over infinite index sets.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Identity Laws
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free