Let α,β be the roots of x2−x+p=0andγ,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
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 x2−x+p=0 ∴α+β=1...(i)αβ=p...(ii) γδ 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(v)⇒a=11+r=11+2or11−2=13or−1 As p is an integer (given), r is also an integer (2 or -2). ∴(vi)⇒a≠13. Hence a=−1andr=−2.∴p=(−1)2×(−2)=−2q=(−1)2×(−2)5=−32