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x2 − x + 1 = 0 αβ = 1 α + β =1 cubing on both sides α3 + β3 + 3αβ(α + β) = 1 α3 + β3 + 3 ∗ 1(1 ) = 1 α3 + β3 = -2 α3 β3 = 1 Sum of the roots=-2 product=1 Required equation is x2 + 2x + 1 = 0
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The speeds of the three cars are in the ratio of 4: 3: 5. The ratio of respective time taken by these cars to travel the same distance is: