Question

Let $\pi ,\beta ϵR,\alpha {,\beta }^2$ be the roots of quadratic equation $x^2-px=1=0$ and ${\pi }^2,\beta$ be the roots of quadratic equation $x^2-qx=8=0$.If $\frac{r}{8}$ be the arithmetic mean of p and q then the value of r is

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

Answer 1

The correct option is A $\frac{83}{8}$

$=\frac{p+q}{2}=\frac{83}{8}$
${\alpha }^2+\beta =q;{\alpha }^2\beta =8$
$\alpha +{\beta }^2+{\alpha }^2+\beta =p+q$
$(\alpha +\beta )+(\alpha +\beta {)}^2-4=p=q$
$\alpha =4=\frac{1}{2}$
$(4+\frac{1}{2})+4+\frac{1}{2}{)}^2-4-p+q$
$\frac{9}{2}+\frac{81}{y}=p+q$
$\frac{81+18-16}{4}=p+q$
$\frac{83}{4}=p+q$
${\alpha }^3{\beta }^3=8$
$\alpha \beta =2$
$frac{\alpha }^2\beta {\alpha }^2\beta =8$
$\frac{\alpha }{\beta }=8$
$\alpha =8\beta$
$\therefore 8{\beta }^2=2$
${\beta }^2=\frac{1}{4}$
$\beta =\pm \frac{1}{2}$
$\alpha =\pm 4$