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Mathematics Question on Relations and functions

On the set of positive rationals, a binary operation * is defined by a * b = 2ab5\frac{2ab}{5} . If 2 * x = 3−13^{-1} then x =

A

25\frac{2}{5}

B

16\frac{1}{6}

C

12548\frac{125}{48}

D

512\frac{5}{12}

Answer

12548\frac{125}{48}

Explanation

Solution

a∗e=aa^{ *}e=a
a∗e=a⇒ 2ae5=a⇒e=52a^{* }e=a \Rightarrow\, \frac{2 ae}{5}=a \Rightarrow e=\frac{5}{2}
a∗a−1=e⇒2aa−15=52⇒a−1=254aa^{ *}a^{-1}=e\Rightarrow \frac{2a a^{-1}}{5}=\frac{5}{2} \Rightarrow a^{-1}=\frac{25}{4a}
2∗x=3−1⇒2(2x)5=254(3)2^{ *}x=3^{-1}\Rightarrow\frac{2\left(2x\right)}{5}=\frac{25}{4\left(3\right)}
⇒ x=12548\Rightarrow\, x=\frac{125}{48}