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Question
Let * be a binary operation on Q+ defined by \[a * b = \frac{ab}{100} \text{ for all a, b } \in Q^+\] The inverse of 0.1 is _________________ .
Options
105
104
106
none of these
Solution
105
Let e be the identity element in Q+with respect to * such that
\[a * e = a = e * a, \forall a \in Q^+ \]
\[a * e = a \text{ &}e * a = a, \forall a \in Q^+ \]
\[\frac{ae}{100} = a \text{ & }\frac{ea}{100} = a, \forall a \in Q^+ \]
\[e = 100 , \forall a \in Q^+\]
Thus, 100 is the identity element in Q+ with respect to *.
\[\text{ Let } b \in Q^+\text{ be the inverse of 0.1.Then }, \]
\[0.1 * b = e = b * 0 . 1\]
\[0 . 1 * b = e \text{ and }b * 0 . 1 = e\]
\[\frac{0 . 1b}{100} = 100 \text{ and }\frac{b\left( 0 . 1 \right)}{100} = 100\]
\[b = \frac{100 \times 100}{0 . 1}\]
\[ = {10}^5 \in Q^+ \]
\[\text{ Thus }, {10}^5 \text{ is the inverse of } 0 . 1 . \]
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