QJEEM24Ma193b: JEE Main 2024 Mathematics - Class 11 JEEM24Ma193b Solution: If , where denotes the greatest integer less than or equal to…

$\begin{aligned}& S:\{a \in R:|2 a-1|=3[a]+2\{a\}\} \\& |2 a-1|=3[a]+2(a-[a]) \\& |2 a-1|=[a]+2 a\end{aligned}$

Case I: If 0 < a < \frac{1}{2}

$\begin{aligned}& 1-2 a=0+2 a \\& \Rightarrow a=\frac{1}{4}\end{aligned}$

Case II: If \frac{1}{2} < a < 1

$2 a-1=0+2 a$

No solution

Case III: If 1 \leq a<2

$2 a-1=1+2 a$

$\Rightarrow$ No solution

$\therefore$ only solution is $a=\frac{1}{4}$

$$72\sum _{a\in S}a=72\times \frac{1}{4}=18$$