If both $(x-2)$ and $\left(x-\frac{1}{2}\right)$ are factors of $p x^{2}+5 x+r$,

Solution:

Let $f(x)=p x^{2}+5 x+r$

It is given that $(x-2)$ is a factor of $f(x)$.

Using factor theorem, we have

$f(2)=0$

$\Rightarrow p \times\left(\frac{1}{2}\right)^{2}+5 \times \frac{1}{2}+r=0$

$\Rightarrow \frac{p}{4}+r=-\frac{5}{2}$

$\Rightarrow p+4 r=-10 \quad \ldots \ldots(2)$

From (1) and (2), we have

$4 p+r=p+4 r$

$\Rightarrow 4 p-p=4 r-r$

$\Rightarrow 3 p=3 r$

$\Rightarrow p=r$