Question:
Let \( * \) be defined on \( \mathbb{Q}^+ \) by:
\[ a * b = \frac{ab}{3} \]
Find the inverse of \( 4 * 6 \).
Options:
- (a) \( \frac{9}{8} \)
- (b) \( \frac{2}{3} \)
- (c) \( \frac{3}{2} \)
- (d) None of these
Solution:
Step 1: Find identity element
Let identity be \( e \), then:
\[ \frac{ae}{3} = a \Rightarrow ae = 3a \Rightarrow e = 3 \]
Step 2: Compute \( 4 * 6 \)
\[ 4 * 6 = \frac{4 \cdot 6}{3} = \frac{24}{3} = 8 \]
Step 3: Find inverse of 8
Let inverse be \( x \), then:
\[ 8 * x = 3 \]
\[ \frac{8x}{3} = 3 \Rightarrow 8x = 9 \Rightarrow x = \frac{9}{8} \]
—Final Answer:
\[ \boxed{\frac{9}{8}} \]
Correct Option: (a)