1. Let α, β be the roots of the equation x2 - px + r = 0 and α/2, 2β be the roots of the equation x2 - qx + r = 0. Then the value of r is ?

Answer 1

"Let α, β be the roots of the equation x2 - px + r = 0"
This implies:
(x-α)(x-β)=0
or
r=αβ ...(1)
p=α+β ...(2)

"and α/2, 2β be the roots of the equation x2 - qx + r = 0."
implies:
(x-α/2)(x-2β)=0
or
r=αβ ...(1a)
q=α/2+2β ...(3)

From (2) and (3) solve for α and β in terms of p and q, hence find r=αβ