Question

Let $\alpha, \beta$ are the roots of $375 x ^{2}-25 x -2=0$ and $y=\displaystyle\sum_{ r =1}^{10}( t - r )^{2}$, where $t \in R$
If $a, b$ and $c$ also represent the sides of a triangle, then the complete set of $\alpha^{2}$ is

• A. $\left(\frac{1}{3}, 3\right)$
• B. $(2,3)$
• C. $\left[\frac{1}{3}, 2\right]$
• D. $\left(\frac{\sqrt{5}+3}{2}, 3\right)$

Answer 1

Answer: (D)

Let $b=a r, c=a r^{2}$, and $r>0$
As the sum of two sides is more than the third side, we have
$r \in\left(\frac{\sqrt{5}-1}{2}, \frac{\sqrt{5}+1}{2}\right)-\{1\}$
$\Rightarrow r+\frac{1}{r}-1 \in(1, \sqrt{5}-1)$
As $\alpha^{2}=\frac{r^{2}+r+1}{r^{2}-r+1}=1+\frac{2}{r+\frac{1}{r}-1}$
$\therefore \alpha^{2} \in\left(\frac{\sqrt{5}+3}{2}, 3\right)$