Further Mathematics 7366: Complex Numbers Topical Past Paper Questions

The circle C represents the locus of points satisfying the equation $$ |z-a\mathrm{i}|=b $$ where $a$ and $b$ are real constants. The circle C intersects the imaginary axis at 2i and 8i The circle C is shown on the Argand diagram in Figure 2 Figure 2 (a) (i) Write down the value…

(a) Show that $ (1 + i)^{4} = -4 $ [3 marks] (b) The function f is defined by $$ \mathrm{f}(z)=z^{4}+3z^{2}-6z+10\quad z\in\mathbb{C} $$ (b) (i) Show that $ (1 + i) $ is a root of f(z) = 0 [2 marks] (b) (ii) Hence write down another root of f(z) = 0 [1 mark] (b) (iii) One of the…

Sketch, on the Argand diagram below, the locus of points satisfying the equation $$ \left|z-2\mathrm{i}\right|=2 $$ [2 marks] (b) Sketch, also on the Argand diagram above, the locus of points satisfying the equation $$ \arg z=\frac{\pi}{3} $$ [1 mark] For the complex number w fin…