If the roots of ` x^3 - 9x^2 +kx +l=0` are in A.P with common difference 2 then (k,l) =

Options

A. `(15-15)`
B. `(23,-15)`
C. `(15,-23)`
D. `(-15 ,23)`

Answer

To solve the problem, we need to find the values of \( k \) and \( l \) in the cubic equation \( x^3 - 9x^2 + kx + l = 0 \) given that its roots are in arithmetic progression (A.P.) with a common difference of 2.

### Step-by-step Solution:

1. **Define the Roots**:
Since the roots are in A.P. with a common difference of 2, we can denote the roots as:
\[ a - 2, \quad a, \quad a + 2 \]

2. **Sum of the Roots**:
According to Vieta's formulas, the sum of the roots of the polynomial \( x^3 - 9x^2 + kx + l = 0 \) is equal to the coefficient of \( x^2 \) with a negative sign. Therefore:
\[ (a - 2) + a + (a + 2) = -(-9) = 9 \]
Simplifying the left side:
\[ 3a = 9 \]
Thus, we find:
\[ a = 3 \]

3. **Finding the Roots**:
Now substituting \( a = 3 \) back into the roots:
\[ \text{Roots are: } 3 - 2 = 1, \quad 3, \quad 3 + 2 = 5 \]
So the roots are \( 1, 3, 5 \).

4. **Forming the Polynomial**:
The polynomial can be formed from its roots as follows:
\[ (x - 1)(x - 3)(x - 5) \]

5. **Expanding the Polynomial**:
First, we expand \( (x - 1)(x - 5) \):
\[ (x - 1)(x - 5) = x^2 - 6x + 5 \]
Now multiply this result by \( (x - 3) \):
\[ (x^2 - 6x + 5)(x - 3) = x^3 - 3x^2 - 6x^2 + 18x + 5x - 15 \]
Combining like terms:
\[ = x^3 - 9x^2 + 23x - 15 \]

6. **Identifying \( k \) and \( l \)**:
From the expanded polynomial \( x^3 - 9x^2 + 23x - 15 \), we can identify:
\[ k = 23, \quad l = -15 \]

### Final Answer:
Thus, the values of \( k \) and \( l \) are:
\[ (k, l) = (23, -15) \]