Let * be a binary operation defined on set `Q-{1}` by the rule `a`*`b=a+b-a b`. Then, the identity element for * is

Options

A. `1`
B. `(a-1)/a`
C. `a/(a-1)`
D. 0

Explanation

To find the identity element for the binary operation defined by \( a * b = a + b - ab \) on the set \( \mathbb{Q} - \{1\} \), we will denote the identity element as \( e \). The identity element must satisfy the condition that for any element \( a \) in the set, the operation with \( e \) yields \( a \).

### Step-by-Step Solution:

1. **Define the Identity Element**:
We start by stating that \( e \) is the identity element if:
\[ a * e = a \]
for all \( a \in \mathbb{Q} - \{1\} \).

2. **Substitute the Operation**:
Using the definition of the operation, we can write:
\[ a * e = a + e - ae \]
Setting this equal to \( a \), we have:
\[ a + e - ae = a \]

3. **Simplify the Equation**:
To isolate \( e \), we can subtract \( a \) from both sides:
\[ e - ae = 0 \]

4. **Factor Out \( e \)**:
We can factor \( e \) out of the left-hand side:
\[ e(1 - a) = 0 \]

5. **Solve for \( e \)**:
The equation \( e(1 - a) = 0 \) implies that either \( e = 0 \) or \( 1 - a = 0 \). Since \( a \) can be any element in \( \mathbb{Q} - \{1\} \), the only consistent solution is:
\[ e = 0 \]

6. **Verify the Identity Element**:
To confirm that \( e = 0 \) is indeed the identity element, we check:
\[ a * 0 = a + 0 - a \cdot 0 = a \]
and
\[ 0 * a = 0 + a - 0 \cdot a = a \]
Thus, both \( a * 0 = a \) and \( 0 * a = a \) hold true.

### Conclusion:
The identity element for the binary operation \( * \) defined on the set \( \mathbb{Q} - \{1\} \) is:
\[ \boxed{0} \]