(C) Given $f(x, y) = (x+y)(x \omega + y \omega^2)(x \omega^2 + y \omega)$.We know that $(x \omega + y \omega^2)(x \omega^2 + y \omega) = x^2 \omega^3

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(C) Given $f(x, y) = (x+y)(x \omega + y \omega^2)(x \omega^2 + y \omega)$.We know that $(x \omega + y \omega^2)(x \omega^2 + y \omega) = x^2 \omega^3 + xy \omega^2 + xy \omega^4 + y^2 \omega^3 = x^2 + xy(\omega^2 + \omega) + y^2 = x^2 - xy + y^2$ (since $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$).Thus,$f(x, y) = (x+y)(x^2 - xy + y^2) = x^3 + y^3$.Substituting $x = 2$ and $y = 3$:$f(2, 3) = 2^3 + 3^3 = 8 + 27 = 35$.

Class 11 Mathematics 4-1.Complex numbers De Moivre's theorem and Roots of unity

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If $1, \omega, \omega^2$ are the cube roots of unity and $(x+y)(x \omega+y \omega^2)(x \omega^2+y \omega)=f(x, y)$,then $f(2, 3)=$

AP EAMCET 2023 All AP EAMCET

A
$16$
B
$24$
C
$35$
D
$45$

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