Practice section 9 of 9

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]

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

(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]

In an Argand diagram, $P,Q$ represent $z_1=2+3i$ and $z_2=iz_1$.

(i) Find the area of triangle $OPQ$, where $O$ is the origin. [2]

(ii) These numbers are roots of $(z^2+az+b)(z^2+cz+d)=0$, where $a,b,c,d\in\mathbb R$. Find $a,b,c,d$. [4]

The numbers $z_1,z_2$ satisfy $|z_1|=4$, $\arg z_1=\pi/3$, $|z_2|=2$, $\arg z_2=-\pi/6$.

(i) On an Argand diagram with origin $O$, mark their points and deduce the exact value of $|z_1-z_2|$. [3]

(ii) Write down a number $a$ such that $z_1=az_2$. Hence or otherwise describe their geometric relationship. [2]

Given $z=1+i$ and $w=\sqrt3+i$, without the use of a calculator, find the exact value of $\left|\dfrac zw\right|$. [3]

Do not use a calculator. One root of $zz^*+2iz=a+6i$, with $a\in\mathbb R$, is $z=3-7i$. Find $a$ and the other root. [4]

Do not use a calculator.

(a) Show that $z=2i$ is a root of $z^3+2z+4i=0$. [2] Hence find the other roots. [3]

(b) Given $w_1=-1-i$ and $w_2=-\dfrac{\sqrt6}{2}+\dfrac{\sqrt2}{2}i$, find their moduli and arguments exactly. [2]