These Ten-Year-Series (TYS) worked solutions with video explanations for 2022 A Level H2 Mathematics Paper 2 Question 3 are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
2022 A Level H2 Math Paper 2 Question 3
The complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are such that ${{z}_{1}}=3-\mathbf{i}\sqrt{3}$, ${{z}_{2}}=\frac{1}{2}{{\text{e}}^{\mathbf{i}\frac{2\pi }{5}}}$ and ${{z}_{3}}={{z}_{1}}\times {{z}_{2}}$.
(a)
Find exactly the modulus and argument of ${{z}_{3}}$.
[3]
(a) Find exactly the modulus and argument of ${{z}_{3}}$.
[3]
(b)
Sketch an Argand diagram showing ${{z}_{1}}$, ${{z}_{2}}$ and $x$.
[2]
(b) Sketch an Argand diagram showing ${{z}_{1}}$, ${{z}_{2}}$ and $x$.
[2]
(c)
Find the smallest positive integer value of $n$ for which ${{z}_{3}}^{n}$ is purely imaginary. State the modulus and argument of ${{z}_{3}}^{n}$ in this case, giving the modulus in the form $k\sqrt{3}$, where $k$ is an integer.
[4]
(c) Find the smallest positive integer value of $n$ for which ${{z}_{3}}^{n}$ is purely imaginary. State the modulus and argument of ${{z}_{3}}^{n}$ in this case, giving the modulus in the form $k\sqrt{3}$, where $k$ is an integer.
[4]
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