Review practice · Section 2 of 3

A graphic calculator is not to be used in answering this question.

Express $(1-4i)^2$ in the form $a+bi$. Hence find the roots of $\left(\dfrac z2+3\right)^2=15+8i$.4 marks

Do not use a graphic calculator in answering this question.

(a) The diagram shows $y=f(x)$ for a cubic polynomial with real coefficients. What can be said about all roots of $f(x)=0$?2 marks

(b) The equation $2z^2-(7+6i)z+11+ic=0$, where $c$ is a nonzero real number, has a root $z=3+4i$. Show that $c=-2$ and determine the other root in Cartesian form. Hence find the roots of $2w^2+(-6+7i)w-11+2i=0$.6 marks

(a) The complex numbers $z,w$ satisfy $z+w^*+5i=10$ and $|w|^2=z+18+i$. Find $z$ and $w$.4 marks

(b)(i) Given that $2+i$ is a root of $z^2-5z+7+i=0$, find the second root in Cartesian form, showing your working.2 marks

(b)(ii) Hence find the roots of $-iw^2+5w+7i-1=0$.2 marks

Find the roots of $iz^2-(5+i)z+2-6i=0$, giving answers in Cartesian form $a+bi$ where $a,b\in\mathbb R$.2 marks

Without the use of a calculator, find the complex numbers $z,w$ satisfying $z-wi=3$ and $z^2-w+6+3i=0$.6 marks

The equation $z^4-4z^3+az^2-20z+25=0$, $a\in\mathbb R$, has a root $ki$ with $k\in\mathbb R$.

(i) Explain clearly why more than one value of $k$ is possible.1 mark

(ii) Find the possible exact values of $k$, show that $a=10$, and hence find the roots.7 marks

(iii) Deduce the roots in Cartesian form of (a) $w^4+4iw^3-10w^2-20iw+25=0$ and (b) $25v^4-20v^3+10v^2-4v+1=0$.4 marks