If the H.C.F. of the expression `(a^2-1)` and `pa^2 - q (a+1)` is (a-1) then relation between p and q will be

Options

A. p=q
B. p=2q
C. p=2q+1
D. p=q+1

Answer

To solve the problem, we need to find the relationship between \( p \) and \( q \) given that the H.C.F. of the expressions \( (a^2 - 1) \) and \( (pa^2 - q(a + 1)) \) is \( (a - 1) \).

### Step-by-step Solution:

1. **Factor the first expression**:
The expression \( a^2 - 1 \) can be factored as:
\[ a^2 - 1 = (a - 1)(a + 1) \]

2. **Set up the H.C.F. condition**:
We know that the H.C.F. of \( (a^2 - 1) \) and \( (pa^2 - q(a + 1)) \) is \( (a - 1) \). This means that \( (a - 1) \) must divide \( (pa^2 - q(a + 1)) \).

3. **Substitute \( a = 1 \)**:
To find the relationship between \( p \) and \( q \), we can substitute \( a = 1 \) into the expression \( (pa^2 - q(a + 1)) \):
\[ p(1^2) - q(1 + 1) = p - 2q \]

4. **Set the expression to zero**:
Since \( (a - 1) \) is a factor, we set the expression equal to zero:
\[ p - 2q = 0 \]

5. **Solve for \( p \)**:
Rearranging the equation gives us:
\[ p = 2q \]

### Conclusion:
The relationship between \( p \) and \( q \) is:
\[ p = 2q \]