$\begin{array}{l} α, β a r e t h e r e a l r o o t s o f a x^{2} + b x + c = 0 o b s e r v e t h e f o l l o w i n g l i s t s \\ L i s t - I L i s t - I I \\ A) a, b, c h a v e s a m e s i g n 1) α, β = 1 \\ B) I f a = c 2) - 1 \\ C) I f a + b + c = 0 t h e n o n e r o o t i s 3) α a n d β a r e n e g a t i v e \\ D) I f a + c = b t h e n o n e r o o t i s 4) 0 \\ 5) 1 \\ T h e n t h e c o r r e c t m a t c h i n g i s \end{array}$

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$1 1 2 5$

$1 3 4 5$

$3 1 5 4$

$3 1 5 2$

answer is D.

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Detailed Solution

$\begin{array}{l} I f α, β a r e r o o t s o f a x^{2} + b x + c = 0 t h e n \\ α + β = \frac{- b}{a}, α β = \frac{c}{a} \\ A) I f a, b, c h a v e s a m e s i g n t h e n \\ ∴ α + β < 0 t h e n α β > 0 \\ ∴ α a n d β a r e n e g a t i v e \\ B) I f a = c t h e n α β = 1 \\ \Rightarrow α = β = 1 \\ C) I f a + b + c = 0 t h e n x = 1 i s r o o t \\ D) I f a + c = b \Rightarrow a - b + c = 0 \\ t h e n x = - 1 i s r o o t \\ ∴ A \to 3, B \to 1, C \to 5, D \to 2 \end{array}$