InterviewSolution
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InterviewSolution
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The real numbers `x_1, x_2, x_3`satisfying the equation `x^3-x^2+b x+gamma=0`ar ein A.P. Find the intervals in which `betaa n dgamma`lie.A. ` (- (1)/(9), infty)`B. `(-(1)/(27) , + infty)`C. `((2)/(9), + infty)`D. none of these |
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Answer» Correct Answer - 2 From the question, the real roots of `x^(3) - x^(2) + betax + y = 0` are `x_(1),x_(2),x_(3)` and the they are in A.P.As `x_(1),x_(2),x_(3)` are in A.P., let `x_(1) = a-d, x_(2) = a, x_(3) = a + d`. Now, `x_(1) + x_(2) + x_(3) = -(-1)/(1) = 1` `rArr a-d + a+d = 1` ` rArr a= (1)/(3)" "(1)` `x_(1),x_(2)+x_(x_(2)x_(3) + x_(3)+x_(1) = (beta)/(1) = beta` `rArr (a-d)a+a(a+d) + (a+d) (a-d) = beta" "(2)` `x_(1)x_(2)x_(3)= -(gamma)/(1) = gamma` `rArr (a-d)a(a+d) = - gamma` Form (1) and (2), we get `3a^(2) -d^(2) = beta ` `rArr 3 (1)/(9) - d^(2) = beta , so beta = (1)/(3) - d^(2) lt (1)/(3)` From (1) and (3) , we get `(1)/(3) ((1)/(9) - d^(2)) = - gamma` `rArr gamma = (1)/(3) ( d^(2)-(1)/(9)) gt (1)/(3) (-(1)/(9)) = - (1)/(27)` ` gamma in (-(1)/(27), + infty)` |
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