If the sum of two roots of the equation `x^3-px^2 + qx-r =0` is zero, then:

Options

A. pq = r
B. qr = p
C. pr = q
D. pqr = 1

Explanation

To solve the problem, we need to analyze the cubic equation given by \( x^3 - px^2 + qx - r = 0 \) and the condition that the sum of two of its roots is zero. Let's denote the roots of the equation as \( \alpha, \beta, \) and \( \gamma \).

### Step-by-Step Solution:

1. **Identify the Roots**:
We know that the roots of the cubic equation are \( \alpha, \beta, \) and \( \gamma \). Given that the sum of two roots is zero, we can assume:
\[ \alpha + \beta = 0 \]

2. **Express One Root in Terms of the Other**:
From the equation \( \alpha + \beta = 0 \), we can express \( \beta \) in terms of \( \alpha \):
\[ \beta = -\alpha \]

3. **Use the Sum of Roots Formula**:
The sum of the roots of the cubic equation can also be expressed as:
\[ \alpha + \beta + \gamma = p \]
Substituting \( \beta = -\alpha \) into this equation gives:
\[ \alpha - \alpha + \gamma = p \implies \gamma = p \]

4. **Substitute the Root into the Original Equation**:
Since \( \gamma \) is a root of the equation, we can substitute \( \gamma = p \) into the original cubic equation:
\[ p^3 - p(p^2) + qp - r = 0 \]

5. **Simplify the Equation**:
Simplifying the equation yields:
\[ p^3 - p^3 + qp - r = 0 \]
This simplifies to:
\[ qp - r = 0 \]

6. **Rearrange to Find the Relationship**:
Rearranging the equation gives us:
\[ qp = r \]

### Conclusion:
Thus, we have derived that if the sum of two roots of the equation \( x^3 - px^2 + qx - r = 0 \) is zero, then the relationship between the coefficients is:
\[ qp = r \]