Question

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

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

Answer 1

Answer: (D)

For the equation $x ^2- px +1=0$,
the product of roots, $\alpha \beta^2=1$
and for the equation $x ^2- qx +8=0$,
the product of roots $\alpha^2 \beta=8$
Hence, $\left(\alpha \beta^2\right)\left(\alpha^2 \beta\right)=8 \Rightarrow \alpha^3 \beta^3=8 \Rightarrow \alpha \beta=2$
$\therefore $ From $\alpha \beta^2=1$, we have $\beta=\frac{1}{2}$ and from $\alpha^2 \cdot \beta=8$, we have $\alpha=4$
Hence, from sum of roots $=-\frac{b}{a}$ " relation, we have $p=\alpha+\beta^2=4+\frac{1}{4}=\frac{17}{4}$
and $q=\alpha^2+\beta=16+\frac{1}{2}=\frac{33}{2}$
$\because \frac{ r }{8}$ is arithmetic mean of $p$ and $q$
$\therefore \frac{ r }{8}=\frac{ p + q }{2} \Rightarrow r =4( p + q )=4\left(\frac{17}{4}+\frac{33}{2}\right)=17+66=83$