Sets, Relations & Functions
General
Grade 11
Question:
<p>If <span class="math-inline">\(\{x\}=\frac{2}{3}\)</span> and the nested floor/fractional-part expression equals 5, which are correct?<br>(A) <span class="math-inline">\(x=\frac{14}{3}\)</span> (B) <span class="math-inline">\([x]=5\)</span> (C) <span class="math-inline">\(x=\frac{17}{3}\)</span> (D) <span class="math-inline">\([x]=4\)</span></p>
A,B
<strong>A,D</strong>
B,C
C,D
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Let x=n+2/3. The nested expression reduces to [x+2/3]=5, giving n+4/3+2/3=[n+2]=n+2... wait: [x+2/3]=[n+4/3]=n+1=5 â n=4.</p><p>So x=4+2/3=14/3. [x]=4.</p><p><strong>Answer: (A),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> Fractional part stays 2/3 after adding integers. Don't lose track.</div><div class="key-concept"><strong>Key Concept:</strong> Nested [ ] and { } â reduce to x=n+r with 0âĪr<1</div></div>
Correct Answer: A,D