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Question

If α,β are the roots of the quadratic equation x2+px+q=0, then the values of α3+β3 and α4+α2β2+β4 are respectively

A
3pqp3 and p43p2q+3q2
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B
p(3qp2) and (p2q)(p2+3q)
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C
pq4 and p4q4
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D
3pqp3 and (p2q)(p23q)
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Solution

The correct option is D 3pqp3 and (p2q)(p23q)
Sum of roots, α+β=p and αβ=q

(α3+β3)=(α+β)33αβ(α+β)

=(p)33q(p)

=p3+3pq

and α4+α2β2+β4=(α4β4)+(αβ)2

=(α2+β2)2(αβ)2

=[(α+β)22αβ]2(αβ)2

=[(p)22q]23(q)2

=[p22q]23q2

=p44p2q+4q2q2

=p44p2q+3q2

=p43p2qp2q+3q2

=p2(p23q)q(p23q)

=(p2q)(p23q)

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