`Q^+` denote the set of all positive rational numbers. If the binary operation `o.` on `Q^+` is defined as `a o.b=(a b)/2,` then the inverse of `3` is

Options

A. A. `4/3`
B. B. `1/3`
C. C. `2/3`
D. D. `5/3`

Explanation

To find the inverse of 3 under the binary operation defined on the set of positive rational numbers \( Q^+ \), we follow these steps:

### Step 1: Understand the Binary Operation
The binary operation \( o \) is defined as:
\[ a \, o \, b = \frac{a \cdot b}{2} \]
for \( a, b \in Q^+ \).

### Step 2: Identify the Identity Element
To find the inverse of a number, we first need to identify the identity element \( E \) for this operation. The identity element satisfies the equation:
\[ a \, o \, E = a \]
Substituting the definition of the operation, we have:
\[ \frac{a \cdot E}{2} = a \]
Multiplying both sides by 2 gives:
\[ a \cdot E = 2a \]
Dividing both sides by \( a \) (since \( a \) is positive and non-zero):
\[ E = 2 \]
Thus, the identity element is \( E = 2 \).

### Step 3: Set Up the Inverse Equation
The inverse of a number \( x \) under the operation \( o \) is defined such that:
\[ x \, o \, x^{-1} = E \]
For our case, we want to find the inverse of 3, so we set up the equation:
\[ 3 \, o \, x^{-1} = 2 \]

### Step 4: Substitute the Operation Definition
Using the definition of the operation, we substitute:
\[ \frac{3 \cdot x^{-1}}{2} = 2 \]

### Step 5: Solve for \( x^{-1} \)
To eliminate the fraction, multiply both sides by 2:
\[ 3 \cdot x^{-1} = 4 \]
Now, divide both sides by 3:
\[ x^{-1} = \frac{4}{3} \]

### Conclusion
The inverse of 3 under the operation \( o \) is:
\[ \boxed{\frac{4}{3}} \]