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Question #41699

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all ordered pairs $(x, y)$ of real numbers satisfying the system of equations: $\binom{8}{[x]} + \binom{8}{[y]} = 84$ $[x + y] = [x] + [y]$ $x - y = [x] - [y] - \frac{1}{5}$ $2x + 3y = 2[x] + 3[y] + \frac{41}{10}$ where $[t]$ denotes the greatest integer less than or equal to $t$ . Find the number of elements in $S$ .

Options

A.
0
B.
4
C.
8
D.
16

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

Inconsistency Detection|Integral Part and Fractional Part|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the candidate's ability to detect systemic inconsistencies hidden behind algorithmic distractors.

The trap is set by providing a solvable binomial equation that yields exactly 8 integral pairs, tempting students to immediately conclude there are 8 solutions.

However, the linear equations for the fractional parts yield $\{x\} = 0.7$ and $\{y\} = 0.9$ .

The critical Advanced-level disqualifier is the step function property $[x+y] = [x] + [y] + [\{x\} + \{y\}]$ .

Since $0.7 + 0.9 \ge 1$ , this forces $[x+y] = [x] + [y] + 1$ , which directly violates the explicit constraint $[x+y] = [x] + [y]$ , rendering the entire feasible region a void zone.

Quick Facts About This Question

Question Information

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