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Question

Let p,q be integers and let α,β be the roots of the equation, x2x1=0, where αβ. For n=0,1,2,...., let an=pαn+qβn.

FACT: If a and b are rational numbers and a+b5=0. then a=0=b.

If a4=28, then p+2q=

A
21
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B
14
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C
7
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D
12
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Solution

The correct option is D 12
α,β are the roots of the equation x2x1=0
α+β=1α=1β (1)
Also, α2α1=0α2=α+1
α4=(α+1)2=α2+2α+1=3α+2
Similarly, β4=3β+2

Now, a4=pα4+qβ4=28
p(3α+2)+q(3β+2)=28
p(3α+2)+q(53α)=28 [From (1)]α(3p3q)+2p+5q=28
p=q and 2p+5q=28 [Using the given fact]
p=q=4
p+2q=4+2×4=12

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