Theory of Equations 1 Question 21

22. Let p and q be real numbers such that p0,p3q and p3q. If α and β are non-zero complex numbers satisfying α+β=p and α3+β3=q, then a quadratic equation having αβ and βα as its roots is

(2010)

(a) (p3+q)x2(p3+2q)x+(p3+q)=0

(b) (p3+q)x2(p32q)x+(p3+q)=0

(c) (p3q)x2(5p32q)x+(p3q)=0

(d) (p3q)x2(5p3+2q)x+(p3q)=0

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Solution:

  1. Sum of roots =α2+β2αβ and product =1

Given, α+β=p and α3+β3=q

(α+β)(α2αβ+β2)=qα2+β2αβ=qp

and

(α+β)2=p2

α2+β2+2αβ=p2

From Eqs. (i) and (ii), we get

α2+β2=p32q3p and αβ=p3+q3p

Required equation is, x2(p32q)x(p3+q)+1=0

(p3+q)x2(p32q)x+(p3+q)=0



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