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Let $α, β$ be two distinct solutions of the equation $[x^{2}] - 3 {x + 1} + 2 = 0,$ where $[x]$ and ${x}$ denotes G.I.F. and F.P.F. respectively. Then the quadratic equation with roots $\frac{α}{β}$ and $\frac{β}{α}$ is

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$2 x^{2} + 5 x + 2 = 0$

$2 x^{2} - 5 x + 2 = 0$

$6 x^{2} + 13 x + 6 = 0$

$6 x^{2} - 13 x + 6 = 0$

answer is A.

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Detailed Solution

$\begin{array}{l} [x^{2}] + 2 = 3 {x} < 3 \\ ∴ [x^{2}] = 0 \Rightarrow x \in (- 1, 1) and {x} = \frac{2}{3} \\ ∴ x = - 1 + \frac{2}{3} or \frac{2}{3} \end{array}$

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