If where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then is equal to

Question

If $S = {a \in R : | 2 a - 1 | = 3 [a] + 2 {a}}$ , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72 \sum_{a \in S} a$ is equal to ________ [2024]

Smart Achievers · Accepted Answer

(18)

$We have, S = {a \in ℝ : | 2 a - 1 | = 3 [a] + 2 {a}}$

$Since for x \in ℝ, x = [x] + {x}$

$∴ 3 [a] + 2 {a} = 2 a + [a]$

$∴ | 2 a - 1 | = 2 a + [a]$

$For 2 a - 1 < 0, we have$

$- 2 a + 1 = 2 a + [a]$

$\Rightarrow 4 a = 1 - [a] \Rightarrow 4 a + [a] - 1 = 0$

$\Rightarrow a = \frac{1}{4}$

$Now, if 2 a - 1 > 0 \Rightarrow a > \frac{1}{2}$

$\Rightarrow 2 a - 1 = 2 a + [a]$

$\Rightarrow [a] = - 1 which is not possible as a > \frac{1}{2}$

$S o, a = \frac{1}{4} \in S$

$∴ 72 \sum_{a \in S} a = 72 \times \frac{1}{4} = 18$

Solution

Q.
If $S = {a \in R : | 2 a - 1 | = 3 [a] + 2 {a}}$ , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72 \sum_{a \in S} a$ is equal to ________ [2024]

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Solution

Q. If S={a∈R:|2a-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∑a∈Sa is equal to ________ [2024]

Ans. (18) We have, S={a∈ℝ:|2a-1|=3[a]+2{a}} Since for x∈ℝ, x=[x]+{x} ∴ 3[a]+2{a}=2a+[a] ∴ |2a-1|=2a+[a] For 2a-1<0, we have -2a+1=2a+[a] ⇒4a=1-[a]⇒4a+[a]-1=0 ⇒a=14 Now, if 2a-1>0⇒a>12 ⇒ 2a-1=2a+[a] ⇒[a]=-1 which is not possible as a>12 So, a=14∈S ∴ 72∑a∈Sa=72×14=18

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Q.
If $S = {a \in R : | 2 a - 1 | = 3 [a] + 2 {a}}$ , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72 \sum_{a \in S} a$ is equal to ________ [2024]