Review practice · Section 1 of 3

Find the possible values of $z$ in the form $x+iy$, where $x,y\in\mathbb R$, satisfying both $\dfrac{z}{z^*}=\dfrac35+\dfrac45i$ and $zz^*=5$.4 marks

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

Two complex numbers are $p=1-i$ and $q=-1+\sqrt3\,i$, and $z=p/q$. Express $z$ in the form $x+iy$, where $x,y$ are exact real values to be determined.2 marks

The complex numbers $w,z$ satisfy $\dfrac wz=\dfrac32i$ and $z^*-2w=4+4i$. Find $w$ and $z$.4 marks

The complex numbers $z_1=2$ and $z_2=i$ are roots of $z^3+Az^2+Bz+C=0$, where $A,B,C$ are real.

(i) State the third root and find $A,B,C$.3 marks

(ii) Hence, with a suitable substitution, find the roots of $2w^3-w^2+2w-1=0$.3 marks

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

One root of $x^4+ax^3+5x^2-x-10=0$, where $a$ is real, is $x=1+2i$. Find $a$ and the other roots.5 marks

Hence find the $x$-coordinates of the intersection point(s) of $y=5(x^2-2x^4)$ and $y=x^3+3x-1$.3 marks

Express $(3-i)^2$ in the form $a+ib$. Hence or otherwise find the roots of $(z+i)^2=-8+6i$.4 marks