IIT JEE Advanced 2007 Paper 1

Since α and β are the roots of equation x2 – px + r = 0, we get
α + β = p (1)
αβ = r (2)
Also, since α / 2 and 2β are the roots of equation x2 – qx + r = 0,
we get

α

2

+2β = q
⇒ α + 4β = 2q (3)
αβ = r
Solving Eqs. (1) and (3), we get
α =

2(2p−q)

3

and β =

2q−p

3

Substituting α and β in Eq. (3), we get
r =

2

9

(2p - q) (2q - p)