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If alpha, beta, gamma are the roots fo x^(3)-x^(2)+ax+b=0 and beta, gamma, delta are the roots of x^(3)-4x^(2)+mx+n=0. If alpha, beta, gamma and delta are INAP with common difference d then |
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Answer» `a=m` `beta=alpha +d, gamma =alpha +2X` and `delta=alpha+3d`…i Given `a, beta, gamma` are the roots of `x^(3)-x^(2)+ax+b=0` then `alpha+beta+gamma=1`………….ii `alpha beta+beta gamma+gamma alpha =a`...........iii `alpha beta gamma=-b`...........IV Also `beta, gamma, delta` are the roots of `x^(3)-4X^(2)+mx+n=0` then `beta+gamma+dleta=4`...........v `beta gamma+gamma delta+delta beta=m`.......vi `beta gamma delta=-n`..........vii From eqs i and ii we get `3 alpha +3 d=1`............viii and from Eqs i and v we get `3 alpha +6d=4` .IX From eqs viii and ix we get `d=1, alpha=-2/3` NOw from Eq i we get `beta=1/3, gamma=4/3` and `delta=7/3` From eqs iii, iv, vi and vii we get `a=-2/3, b=8/27,m=13/3,n=-28/27` `:.a=m-5,n=b-a-2` and `b=m+n-3` |
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