Correct Answer - Option 1 : 28x
2 – 407x + 784 = 0
Given:
x2 – 11x + 28 = 0
Concept Used:
Equation whose roots are α2/β and β2/α:
x2 - ( sum of the roots) x + product of the roots = 0
⇒ x2 – (α2/β + β2/α)x + (α2/β × β2/α) = 0
⇒ x2 – (α2/β + β2/α)x + (αβ) = 0
Calculation:
x2 – 11x + 28 = 0
⇒ x2 – 7x – 4x + 28 = 0
⇒ (x – 7)(x - 4) = 0
⇒ x = 4, 7
∴ α = 4 and β = 7
(α + β) = 11
⇒ α3 + β3 + 3αβ(α + β) = 1331
⇒ α3+ β3 + 84(11) = 1331
⇒ α3+ β3 + 924 = 1331
⇒ α3+ β3= 1331 – 924
⇒ α3+ β3 = 407
&, αβ = 28
Sum of roots = α2/β + β2/α = (α3 +β3)/αβ
Equation whose roots are α2/β and β2/α:
x2 – (α2/β + β2/α)x + (αβ) = 0
x2 – (α2/β + β2/α)x + (αβ) = 0
⇒ x2 – [(α3 +β3)/αβ]x + αβ = 0
⇒ x2 – (407/28)x + 28 = 0
⇒ 28x2 – 407x + 784 = 0