Mathematics Pure Math 2,3 9709: COMPLEX NUMBERS Topical Past Paper Questions

It is given that $ \frac{2+ai}{a-3i} = \lambda(1-5i) $, where $ a $ and $ \lambda $ are real constants. (a) Show that $ a^2 - 5a + 6 = 0 $. (b) Hence, find the possible values of $a$ and the corresponding values of $\lambda$.

Find the complex numbers, $z$, which satisfy the equation $$zz^* + 5iz + 2 - 10i = 0$$ Give your answers in the form $x + iy$, where $x$ and $y$ are real. [5]

Find the complex number $z$ satisfying the equation $$ \frac{z-3\mathrm{i}}{z+3\mathrm{i}}=\frac{2-9\mathrm{i}}{5}. $$ Give your answer in the form $x + \mathrm{i}y$, where $x$ and $y$ are real. [5]