Correct Answer - Option 3 : 1
Concept:
Consider a quadratic equation: ax2 + bx + c = 0.
Let, α and β are the roots.
- Sum of roots = α + β = -b/a
- Product of the roots = α × β = c/a
Cube Roots of unity are 1, ω and ω2
Here, ω = \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\) and ω2 = \(\frac{{ - {\rm{\;}}1{\rm{\;}} - {\rm{\;\;i}}\sqrt 3 }}{2}\)
Property of cube roots of unity
- ω3 = 1
- 1 + ω + ω2 = 0
- ω = 1 / ω 2 and ω2 = 1 / ω
- ω3n = 1
Calculation:
Here, α and β are the roots of the equation x2 - x + 1 = 0
Sum of roots = α + β = 1
Product of roots = α × β = 1
So, α = \(\frac{{ {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\) = -ω2 and β = \(\frac{{ {\rm{\;}}1{\rm{\;}} - {\rm{\;\;i}}\sqrt 3 }}{2}\) = -ω
Now, α3 = (-ω2)3 = -ω6 = -1
β3 = (-ω)3 = -ω3 = -1
α2021 + β2009 = \(\rm \frac{\alpha^{2022} }{\alpha}\rm +\frac{\beta ^{2010}}{\beta}\)
= \(\rm \frac{(\alpha^3)^{674} }{\alpha}\rm +\frac{(\beta ^3)^{670}}{\beta}\)
= \(\rm \frac{(-1)^{674} }{\alpha}\rm +\frac{(-1)^{670}}{\beta}\)
= \(\rm \frac{1 }{\alpha}\rm +\frac{1}{\beta}=\frac{\alpha+\beta }{\alpha\beta}\)
= 1
Hence, option (3) is correct.