$T h e v a l u e o f' a' f o r w h i c h o n e r o o t o f t h e e q u a t i o n (a^{2} - 5 a + 3) x^{2} + (3 a - 1) x + 2 = 0 i s t w i c e a s l \arg e a s t h e o t h e r i s$

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$- \frac{2}{3}$

$\frac{1}{3}$

$- \frac{1}{3}$

$\frac{2}{3}$

answer is D.

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Detailed Solution

$\begin{array}{l} L e t α, 2 α a r e r o o t s o f (a^{2} - 5 a + 3) x^{2} + (3 a - 1) x + 2 = 0 \\ ∴ α + 2 α = \frac{- (3 a - 1)}{a^{2} - 5 a + 3} \\ \Rightarrow 3 α = \frac{1 - 3 a}{a^{2} - 5 a + 3} \Rightarrow α = \frac{1 - 3 a}{3 (a^{2} - 5 a + 3)} \\ a n d α (2 α) = \frac{2}{a^{2} - 5 a + 3} \\ \Rightarrow 2 (\frac{{(1 - 3 a)}^{2}}{9 {(a^{2} - 5 a + 3)}^{2}}) = \frac{2}{a^{2} - 5 a + 3} \\ \Rightarrow 9 (a^{2} - 5 a + 3) = {(1 - 3 a)}^{2} \\ \Rightarrow 9 a^{2} - 45 a + 27 = 1 + 9 a^{2} - 6 a \\ \Rightarrow 26 = 39 a \\ \Rightarrow a = \frac{26}{39} \Rightarrow \frac{2}{3} \end{array}$