1.

If zeros of `x^3 - 3p x^2 + qx - r` are in A.P., then

Answer» Let `alpha-beta, alpha` and `alpha+beta` are the zeroes of given polynomial .
`:.` Sum of zeroes `=-((-3p))/(1)=3p`
i.e., `alpha-beta+alpha+alpha+beta=3p rArr 3 alpha = 3p`
`:. alpha=p " " ...(1)`
Now, since `alpha` is one zero of the polynomial.
`:.` On putting `x=alpha`, we get the remainder=0.
`:. alpha^(3)-3palpha^(2)+qalpha-r=0`
`rArr p^(3)-3p(p^(2))+qp-r=0 " "` [from (1)]
`rArr p^(3)-3p^(3)+pq-r=0`
`rArr 2p^(3)=pq-r`, which is the required relation in p, q and r.


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