If the roots of the equation `x^(3) - px^(2) + qx - r = 0` are in A.P., then

Options

A. `2p^(3) = 9pq - 27r`
B. `2q^(3) = 9pq - 27 r`
C. `p^(3) = 9 pq - 27r`
D. `2p^(3) = 9 pq + 27r`

Explanation

To solve the problem, we need to find the condition that must hold if the roots of the cubic equation \( x^3 - px^2 + qx - r = 0 \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Define the Roots**:
Let the roots of the cubic equation be \( a - d, a, a + d \), where \( a \) is the middle term and \( d \) is the common difference.

2. **Sum of the Roots**:
The sum of the roots can be expressed as:
\[ (a - d) + a + (a + d) = 3a \]
According to Vieta's formulas, the sum of the roots is equal to the coefficient of \( x^2 \) with a negative sign:
\[ 3a = p \quad \text{(1)} \]
From this, we can express \( a \) in terms of \( p \):
\[ a = \frac{p}{3} \]

3. **Substituting the Roots into the Equation**:
Since \( a \) is a root of the equation, we substitute \( x = a \) into the cubic equation:
\[ a^3 - pa^2 + qa - r = 0 \]

4. **Substituting \( a = \frac{p}{3} \)**:
Now, substitute \( a = \frac{p}{3} \) into the equation:
\[ \left(\frac{p}{3}\right)^3 - p\left(\frac{p}{3}\right)^2 + q\left(\frac{p}{3}\right) - r = 0 \]

5. **Simplifying the Equation**:
Calculate each term:
- \( \left(\frac{p}{3}\right)^3 = \frac{p^3}{27} \)
- \( p\left(\frac{p}{3}\right)^2 = p\cdot\frac{p^2}{9} = \frac{p^3}{9} \)
- \( q\left(\frac{p}{3}\right) = \frac{qp}{3} \)

Substitute these into the equation:
\[ \frac{p^3}{27} - \frac{p^3}{9} + \frac{qp}{3} - r = 0 \]

6. **Finding a Common Denominator**:
The common denominator for the fractions is 27. Rewrite the equation:
\[ \frac{p^3}{27} - \frac{3p^3}{27} + \frac{9qp}{27} - r = 0 \]
This simplifies to:
\[ \frac{-2p^3 + 9qp - 27r}{27} = 0 \]

7. **Setting the Numerator to Zero**:
For the equation to hold, the numerator must be zero:
\[ -2p^3 + 9qp - 27r = 0 \]
Rearranging gives us the required condition:
\[ 9qp - 27r = 2p^3 \quad \text{(2)} \]

### Final Condition:
Thus, the condition that must hold if the roots of the equation \( x^3 - px^2 + qx - r = 0 \) are in A.P. is:
\[ 9qp - 27r = 2p^3 \]