If `2` and `(1)/(2)` are the zero of `px^(2)+5x+r`, then :

Options

A. `p=r=2`
B. `p=r=-2`
C. `p=2,r=-2`
D. `p=-2, r=2`

Answer

To find the values of \( p \) and \( r \) in the equation \( px^2 + 5x + r \) given that the zeros are \( 2 \) and \( \frac{1}{2} \), we can follow these steps:

### Step 1: Set up the equation
Since \( 2 \) and \( \frac{1}{2} \) are the roots of the polynomial, we can use Vieta's formulas. The sum of the roots \( (2 + \frac{1}{2}) \) is equal to \( -\frac{b}{a} \) and the product of the roots \( (2 \cdot \frac{1}{2}) \) is equal to \( \frac{c}{a} \).

### Step 2: Calculate the sum of the roots
The sum of the roots is:
\[ 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]
According to Vieta's formulas:
\[ -\frac{5}{p} = \frac{5}{2} \]
Multiplying both sides by \( -p \):
\[ 5 = -\frac{5p}{2} \]
Multiplying both sides by \( -2 \):
\[ -10 = 5p \]
Dividing by \( 5 \):
\[ p = -2 \]

### Step 3: Calculate the product of the roots
The product of the roots is:
\[ 2 \cdot \frac{1}{2} = 1 \]
According to Vieta's formulas:
\[ \frac{r}{p} = 1 \]
Substituting \( p = -2 \):
\[ \frac{r}{-2} = 1 \]
Multiplying both sides by \( -2 \):
\[ r = -2 \]

### Final values
Thus, we have:
\[ p = -2 \quad \text{and} \quad r = -2 \]

### Summary
The values of \( p \) and \( r \) are both \( -2 \).

---