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Question

Let α,β be the roots of x2x+p=0 and γ,δ be the roots of x24x+q=0.~If α,β,γ,δ, are in G.P., then the integral values of p and q respectively, are

A
-2,-32
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B
-2,3
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C
-6,3
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D
-6,-32
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Solution

The correct option is A -2,-32
α,β are the roots of x2x+p=0
α+β=1 ...(i)αβ=p ...(ii)
γδ are the roots of x2+4x+q=0γ+δ=4 ...(iii)γδ=q ...(iv)α,β,γ,δ are in G.P.
Let α=a;β=ar,γ=ar2,δ=ar3.
Substituting these values in equations (i),(ii),(iii) and (iv),
we get
a+ar=1 ...(v)a2r=p ...(vi)ar2+ar3=4 ...(vii)a2r5=q ...(viii)
Dividing (vii) by (v) we get
ar2(1+r)a(1+r)=41r2=4r=2,2(v) a=11+r=11+2 or 112=13or1
As p is an integer (given), r is also an integer (2 or -2).
(vi) a13.
Hence a=1 and r=2.p=(1)2×(2)=2q=(1)2×(2)5=32

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