If p+q=2 , pq=1 then the value of `p^3+q^3+6p^2+5q^2+6pq` is

Options

A. 19
B. 18
C. 15
D. 23

Answer

To solve the problem, we need to find the value of \( p^3 + q^3 + 6p^2 + 5q^2 + 6pq \) given that \( p + q = 2 \) and \( pq = 1 \).

### Step-by-step Solution:

1. **Use the identity for \( p^3 + q^3 \)**:
We know that:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
We can express \( p^2 + q^2 \) using the identity:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]

2. **Calculate \( p^2 + q^2 \)**:
Substitute \( p + q = 2 \) and \( pq = 1 \):
\[ p^2 + q^2 = (2)^2 - 2(1) = 4 - 2 = 2 \]

3. **Calculate \( p^2 - pq + q^2 \)**:
Now we can find \( p^2 - pq + q^2 \):
\[ p^2 - pq + q^2 = (p^2 + q^2) - pq = 2 - 1 = 1 \]

4. **Calculate \( p^3 + q^3 \)**:
Now substitute back into the identity:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) = 2 \cdot 1 = 2 \]

5. **Calculate \( 6p^2 + 5q^2 + 6pq \)**:
We already have \( p^2 + q^2 = 2 \) and \( pq = 1 \):
\[ 6p^2 + 5q^2 + 6pq = 6p^2 + 5q^2 + 6(1) \]
We can express \( 6p^2 + 5q^2 \) as:
\[ 6p^2 + 5q^2 = 6p^2 + 5(2 - p^2) = 6p^2 + 10 - 5p^2 = p^2 + 10 \]

6. **Calculate \( p^2 \)**:
Since \( p + q = 2 \) and \( pq = 1 \), we can find \( p \) and \( q \) using the quadratic equation:
\[ x^2 - (p + q)x + pq = 0 \implies x^2 - 2x + 1 = 0 \]
This simplifies to:
\[ (x - 1)^2 = 0 \implies x = 1 \]
Thus, \( p = 1 \) and \( q = 1 \).

7. **Substituting back**:
Now substituting \( p = 1 \) into \( p^2 \):
\[ p^2 = 1^2 = 1 \]
Therefore:
\[ 6p^2 + 5q^2 + 6pq = 1 + 10 = 11 \]

8. **Final Calculation**:
Now we can find the total:
\[ p^3 + q^3 + 6p^2 + 5q^2 + 6pq = 2 + 11 = 13 \]

### Final Answer:
Thus, the value of \( p^3 + q^3 + 6p^2 + 5q^2 + 6pq \) is \( 13 \).