2021 ACJC P1 Q9
2021 ACJC P1 Q9
11 marks
Prelims
- 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]
- 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]
- Hence find the exact roots of the equation \({{x}^{3}}+3{{x}^{2}}+9x+7-9\mathbf{i}=0\).[2]
- Given that \(z={{\mathrm{e}}^{\mathbf{i}\theta }}\), 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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