JEE PYQ: Quadratic Equation Question 44
Question 44 - 2019 (10 Jan Shift 1)
If 5, $5r$, $5r^2$ are the lengths of the sides of a triangle, then $r$ cannot be equal to:
(1) $\frac{3}{4}$
(2) $\frac{5}{4}$
(3) $\frac{7}{4}$
(4) $\frac{3}{2}$
Show Answer
Answer: (3)
Solution
$\triangle PQR$ is possible if $5 + 5r > 5r^2$
$\Rightarrow 1 + r > r^2$
$\Rightarrow r^2 - r - 1 < 0$
$\Rightarrow \left(r - \frac{1}{2} - \frac{\sqrt{5}}{2}\right)\left(r - \frac{1}{2} + \frac{\sqrt{5}}{2}\right) < 0$
$\Rightarrow r \in \left(\frac{-\sqrt{5}+1}{2}, \frac{\sqrt{5}+1}{2}\right)$
$\therefore \frac{7}{4} \in \left(\frac{-\sqrt{5}+1}{2}, \frac{\sqrt{5}+1}{2}\right)$? No, $\frac{\sqrt{5}+1}{2} \approx 1.618$ and $\frac{7}{4} = 1.75$
$\therefore r \neq \frac{7}{4}$
BETA
