Question

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

• A. $\frac{83}{2}$
• B. 83
• C. $\frac{83}{8}$
• D. $\frac{83}{4}$

Answer 1

Answer: (B)

83

For the equation $x^2-px+1=0$

the product of rotos, $\alpha {\beta }^2=1$

and for the equation $x^2-qx+8=0$,

the product of rotos, ${\alpha }^2\beta =8$

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

$\Rightarrow {\alpha }^2{\beta }^3=8\Rightarrow \alpha \beta =2$

$\therefore$ From $\alpha {\beta }^2=1$, we have $\beta =\frac{1}{2}$ and from ${\alpha }^2.\beta =8$, we have $\alpha =4$

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

$p=\alpha +{\beta }^2=4+\frac{1}{4}=\frac{17}{4}$ and $q={\alpha }^2+\beta =16+\frac{1}{2}=\frac{33}{2}$

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

$\therefore$ $\frac{r}{8}=\frac{p+q}{2}$

$\Rightarrow r=4(p+q)=4(\frac{17}{4}+\frac{33}{2})=17+66=83$