Let $\mathrm{\alpha },\mathrm{\beta }\in \mathrm{R}$. If $\mathrm{\alpha },{\mathrm{\beta }}^2$ are the roots of quadratic equation ${\mathrm{x}}^2-\mathrm{px}+1=0$ and ${\mathrm{\alpha }}^2,\mathrm{\beta }$ is the roots of quadratic equation ${\mathrm{x}}^2-\mathrm{qx}+8=0$, then the value of r if $\frac{\mathrm{r}}{8}$is the arithmetic mean of p and q, is 

For the equation ${\mathrm{x}}^2-\mathrm{px}+1=0$,the product of roots, ${\mathrm{\alpha \beta }}^2=1$

and for the equation ${\mathrm{x}}^2-\mathrm{qx}+8=0$,the product of roots ${\mathrm{\alpha }}^2\mathrm{\beta }=8$

Hence, $\left({\mathrm{\alpha \beta }}^2\right)\left({\mathrm{\alpha }}^2\mathrm{\beta }\right)=8$ $\Rightarrow {\mathrm{\alpha }}^3{\mathrm{\beta }}^3=8\Rightarrow \mathrm{\alpha \beta }=2$

$\therefore$For ${\mathrm{\alpha \beta }}^2=1$, we have $\mathrm{\beta }=\frac{1}{2}$ and from ${\mathrm{\alpha }}^2\cdot \mathrm{\beta }=8$, we have $\mathrm{\alpha }=4$

Hence, from sum of roots $=-\frac{\mathrm{b}}{\mathrm{a}}$, we have

$\mathrm{p}=\mathrm{\alpha }+{\mathrm{\beta }}^2=4+\frac{1}{4}=\frac{17}{4}$

$\text{ and }\mathrm{q}={\mathrm{\alpha }}^2+\mathrm{\beta }=16+\frac{1}{2}=\frac{33}{2}$

$\frac{\mathrm{r}}{8}$ is arithmetic mean of p and q

$\begin{array}{l}\therefore \frac{\mathrm{r}}{8}=\frac{\mathrm{p}+\mathrm{q}}{2}\\ \Rightarrow \mathrm{r}=4(\mathrm{p}+\mathrm{q})=4\left(\frac{17}{4}+\frac{33}{2}\right)=17+66=83\end{array}$