9709 P33 - Nov 2019 - Q6
1951
The complex number with modulus 1 and argument \(\frac{1}{3} \pi\) is denoted by \(w\).
(i) Express \(w\) in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [1]
The complex number \(1 + 2i\) is denoted by \(u\). The complex number \(v\) is such that \(|v| = 2|u|\) and \(\arg v = \arg u + \frac{1}{3} \pi\).
(ii) Sketch an Argand diagram showing the points representing \(u\) and \(v\). [2]
(iii) Explain why \(v\) can be expressed as \(2uw\). Hence find \(v\), giving your answer in the form \(a + ib\), where \(a\) and \(b\) are real and exact. [4]
