If the roots of x3 – 9x2 + 20x – 12 = 0 are in the ratio 1 : 3 then the root is / are

x 3 − 9 x 2 + 20 x − 12 = 0 roots ⁡ α , 3 α , γ α + 3 α + γ = 9 4 α + γ = 9 · · · 1 α . 3 α . γ = 12 α 2 γ = 4 · · · ( 2 ) γ = 4 α 2 4 α + 4 α 2 = 9 ; α = 2 , γ = 1 roots 2, 6, 1

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If the roots of x³ – 9x² + 20x – 12 = 0 are in the ratio 1 : 3 then the root is / are

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answer is A, C, D.

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Detailed Solution

$\begin{array}{l} x^{3} - 9 x^{2} + 20 x - 12 = 0 \\ roots α, 3 α, γ \\ α + 3 α + γ = 9 4 α + γ = 9 \cdot \cdot \cdot (1) \\ α . 3 α . γ = 12 \\ α^{2} γ = 4 \cdot \cdot \cdot (2) \\ γ = \frac{4}{α^{2}} \\ 4 α + \frac{4}{α^{2}} = 9; α = 2, γ = 1 \end{array}$