2021 ACJC P1 Q9
(a)
(i) Show that the cubic polynomial ${{x}^{3}}+p{{x}^{2}}+{{p}^{2}}x+q$ can be reduced to ${{y}^{3}}+\left( \frac{2{{p}^{2}}}{3} \right)y+\alpha $ by the substitution $x=y-\frac{p}{3}$, where $\alpha $ is to be determined in terms of $p$ and $q$.
[3]
(ii) Given that $-3\mathbf{i}$ is a root of the equation ${{y}^{3}}+6y-9\mathbf{i}=0$, find the other two roots exactly in the form $a+b\mathbf{i}$.
[3]
(iii) Hence find the exact roots of the equation ${{x}^{3}}+3{{x}^{2}}+9x+7-9\mathbf{i}=0$.
[2]
(b)
Given that $z={{\text{e}}^{\mathbf{i}\theta }}$, show that $1+z+{{z}^{2}}+{{z}^{3}}+…+{{z}^{n-1}}={{z}^{\frac{n-1}{2}}}\left( \frac{\sin \frac{n\theta }{2}}{\sin \frac{\theta }{2}} \right)$.
[3]
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