If α and β are roots of 3x2 – 6x + 1 = 0, then form a quadratic equation whose roots are
(i) 
(ii) α2β, β2α (iii) 2α + β, 2β + α
3x2 – 6x + 1 = 0 compare this with ax2 – bx + c = 0
∴ a = 3 , b = –6 and c = 1
i). Here 
General form of quadratic equation whose roots are α2 and β2
⇒ 
⇒ 
⇒ 
⇒ 
⇒ x2 – 6x + 3 = 0
Therefore, required equation is x2 – 6x + 3 = 0
ii). Here, α = α2β and β = β2 α
General form of quadratic equation whose roots are α2β and β2 α
x2 – (α2β + β2 α) x + (α2β)(β2 α) = 0
⇒ x2 – αβ (α + β) x + (α3β3) = 0
⇒ x2 – αβ (α + β) x + (αβ)3 = 0
⇒ 
⇒ 
⇒ 
⇒ 27x2 – 18x + 1 = 0
Therefore, required equation is 27x2 – 18x + 1 = 0
iii). Here, α = 2α + β and β = 2β + α
General form of equation whose roots are 2α + β and 2β + α
x2 – (2α + β + 2β + α)x + (2α + β)(2β + α) = 0
x2 – (3α + 3β) x + (4αβ + 2α2 + 2β2 + αβ) = 0
x2 – (3α + 3β) x + (2(α2 + β2 ) + 5αβ) = 0
α2 + β2 = (α + β)2 – 2αβ
x2 – (3α + 3β) x + (2(α2 + β2 ) + 5αβ) = 0
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 3x2 –18x + 70 = 0
Therefore, the required equation is 3x2 –18x + 70 = 0
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