Let α,β be the roots of x2−x+p=0 and γ,δ be the roots of x2−4x+q=0. If α,β,γ,δ, are in G.P., then the integral values of p and q respectively, are
-2,-32
α,β are the roots of x2−x+p=0
∴α+β=1 ...(i)αβ=p ...(ii)
γ,δ are the roots of x2+4x+q=0∴γ+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)=41⇒r2=4⇒r=2,−2
From (v)
⇒a=11+r=11+2 or 11−2=13or−1
As p is an integer (given), r is also an integer (2 or -2).
∴ (vi) ⇒a≠13.
Hence a=−1 and r=−2.∴p=(−1)2×(−2)=−2q=(−1)2×(−2)5=−32