Let `f(x)=x^(2)-ax+b`, `'a'` is odd positive integar and the roots of the equation `f(x)=0` are two distinct prime numbers. If `a+b=35`, then the value of `f(10)=`

Options

A. `-8`
B. `-10`
C. `-4`
D. `0`

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript while providing a clear mathematical approach.

### Step 1: Understand the function and roots
We have the function \( f(x) = x^2 - ax + b \), where \( a \) is an odd positive integer. The roots of this equation are two distinct prime numbers.

### Step 2: Use the properties of roots
From Vieta's formulas, we know:
- The sum of the roots \( P + Q = a \)
- The product of the roots \( P \times Q = b \)

### Step 3: Identify the roots
Since the roots are distinct prime numbers and one of them must be even (to ensure their sum is odd), the only even prime number is 2. Thus, one of the roots is \( P = 2 \). Let the other root be \( Q \), which is also a prime number.

### Step 4: Express \( a \) and \( b \)
From the sum of the roots:
\[ 2 + Q = a \quad \text{(1)} \]
From the product of the roots:
\[ 2 \times Q = b \quad \text{(2)} \]

### Step 5: Use the condition \( a + b = 35 \)
Substituting equations (1) and (2) into the condition \( a + b = 35 \):
\[ (2 + Q) + (2Q) = 35 \]
This simplifies to:
\[ 2 + Q + 2Q = 35 \]
\[ 2 + 3Q = 35 \]
Subtracting 2 from both sides:
\[ 3Q = 33 \]
Dividing by 3:
\[ Q = 11 \]

### Step 6: Find \( a \) and \( b \)
Now substituting \( Q = 11 \) back into equations (1) and (2):
\[ a = 2 + 11 = 13 \]
\[ b = 2 \times 11 = 22 \]

### Step 7: Verify \( a + b = 35 \)
Checking:
\[ a + b = 13 + 22 = 35 \quad \text{(Condition satisfied)} \]

### Step 8: Write the function \( f(x) \)
Now, substituting \( a \) and \( b \) into the function:
\[ f(x) = x^2 - 13x + 22 \]

### Step 9: Calculate \( f(10) \)
Now we need to find \( f(10) \):
\[ f(10) = 10^2 - 13 \times 10 + 22 \]
Calculating each term:
\[ = 100 - 130 + 22 \]
\[ = 100 - 130 = -30 \]
\[ = -30 + 22 = -8 \]

### Final Answer
Thus, the value of \( f(10) \) is:
\[ \boxed{-8} \]