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If `alpha,beta`are the roots of the equation `x^2-2x+3=0`obtain the equation whose roots are `alpha^3-3alpha^2+5alpha-2`and `beta^3-beta^2+beta=5`A. `x^(2) + 3x + 2 = 0`B. `x^(2) - 3x + 2 = 0`C. `x^(2) - 3x - 2 = 0`D. none of these |
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Answer» Correct Answer - B Given `x^(2) - 2x + 3 = 0overset(alpha)underset(beta)(lt)` `alpha + beta = 2` and `alphabeta = 3` `:. alpha^(2) - 2alpha + 3 = 0 rArr alpha^(2) = 2alpha - 3` `alpha^(3) = 2alpha^(2) - 3alpha` Similarly `beta^(3) = 2beta^(2) - 3beta` Now `P = alpha^(3) - 3alpha^(2) + 5alpha - 2` `=2alpha^(2) - 3alpha - 3alpha^(2) + 5alpha - 2 = -alpha^(2) + 2alpha - 2` `= 3 - 2 = 1` `:. Q = beta^(2) - beta^(2) + beta + 5` `= (2beta^(2) - 3beta) - beta^(2) + beta + 5 = beta^(2) - beta + 5` `= -3 + 5 = 2` `:.` Sum of roots `= P + Q = 1 + 2 = 3` `:.` Product `PQ = 2` `:.` Required equation `x^(2) - 3x + 2 = 0`. |
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