9709 P31 - Nov 2009 - Q7
2012
The complex number \(-2 + i\) is denoted by \(u\).
(i) Given that \(u\) is a root of the equation \(x^3 - 11x - k = 0\), where \(k\) is real, find the value of \(k\).
(ii) Write down the other complex root of this equation.
(iii) Find the modulus and argument of \(u\).
(iv) Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities \(|z| < |z - 2|\) and \(0 < \arg(z - u) < \frac{1}{4}\pi\).
