α,β,γ are the roots of the equation x3-10x2+7x+8=0.Match the following.List - I List - II I) α+β+γ a) -434II) α2+β2+γ2 b) -78III) 1α+1β+1γ c) 86IV) αβγ+βγα+γαβ d) 0 e) 10

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$α, β, γ$ are the roots of the equation $x^{3} - 10 x^{2} + 7 x + 8 = 0 .$
Match the following.
List - I List - II

I) $α + β + γ$ a) $- \frac{43}{4}$

II) $α^{2} + β^{2} + γ^{2}$ b) $- \frac{7}{8}$

III) $\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}$ c) 86

IV) $\frac{α}{βγ} + \frac{β}{γα} + \frac{γ}{αβ}$ d) 0

e) 10

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e, c, b, a

d, c, a, b

e, c, a, b

e, b, c, a

answer is C.

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Detailed Solution

$\begin{array}{l} I f α, β, γ a r e r o o t s o f x^{3} - 10 x^{2} + 7 x + 8 = 0 \\ ∴ α + β + γ = 10 \\ \Rightarrow {(α + β + γ)}^{2} = 100 \\ \Rightarrow α^{2} + β^{2} + γ^{2} + 2 (α β + β γ + γ α) = 100 \\ \Rightarrow α^{2} + β^{2} + γ^{2} = 100 - 2 (7) \Rightarrow 100 - 14 = 86 \\ ∴ \frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = \frac{α β + β γ + γ α}{α β γ} \Rightarrow \frac{7}{- 8} \\ \frac{α}{β γ} + \frac{β}{γ α} + \frac{γ}{α β} = \frac{α^{2} + β^{2} + γ^{2}}{α β γ} \Rightarrow \frac{86}{- 8} \Rightarrow \frac{- 43}{4} \end{array}$