Sets, Relations & Functions
Floor and Fractional Part Equation
nta_pyq_2024_apr
Grade 11
Question:
If $S=\left\{a\in\mathbb{R}:|2a-1|=3[a]+2\{a\}\right\}$, where $[t]$ denotes the greatest integer $\leq t$ and $\{t\}$ denotes the fractional part of $t$, then $72\sum_{a\in S}a$ is equal to ________.
Step-by-Step Solution
Key Concept: Note $[2a-1]=3[a]+2\{a\}$. Case $a>1/2$: leads to contradiction. Case $a<1/2$: $a=1/4$.
$a=1/4$. $72\times1/4=18$.
Correct Answer: 18