JEE MAIN - Mathematics (2024 - 5th April Morning Shift - No. 26)
If $$S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a \}\}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$ and $$\{t\}$$ represents the fractional part of $$t$$, then $$72 \sum_\limits{a \in S} a$$ is equal to _________.
Answer
18
Explanation
$$\begin{aligned} & S:\{a \in R:|2 a-1|=3[a]+2\{a\}\} \\ & |2 a-1|=3[a]+2(a-[a]) \\ & |2 a-1|=[a]+2 a \end{aligned}$$
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 $$a=\frac{1}{4}$$
$$72 \sum_\limits{a \in S} a=72 \times \frac{1}{4}=18$$
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