Complex numbers in Cartesian form

Let $z=3+4i$ and $w=1-2i$.

(a) Find $zw$. [2]

(b) Find $\dfrac zw$ in the form $a+bi$. [3]

Let $z=5-2i$.

(a) Simplify $i^{19}$. [1]

(b) Find $z^2$, giving your answer in the form $a+bi$, where $a,b\in\mathbb{R}$. [2]

(c) Find the value of $zz^*$. [1]

The Argand diagram shows the points $A$ and $B$, which represent the complex numbers $z_1$ and $z_2$ respectively.

(a) Write down $z_1$ and $z_2$ in Cartesian form. [1]

(b) Find $z_1-z_2$. [1]

(c) Find $|z_1-z_2|$, giving your answer in the form $p\sqrt{q}$ where $p,q\in\mathbb{Z}$, and state what this value represents on the diagram. [2]

Find the complex number $z$ that satisfies $$2z+iz^*=7+5i,$$ where $z^*$ is the complex conjugate of $z$. Give your answer in the form $x+iy$, where $x,y\in\mathbb{R}$.

Find the two square roots of $-15+8i$, giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$.

Let $z=\dfrac{a+3i}{2-i}$, where $a\in\mathbb{R}$.

(a) Express $z$ in the form $x+iy$, giving $x$ and $y$ in terms of $a$. [3]

(b) Given that $z$ is purely imaginary, find the value of $a$. [2]

(c) Hence write down $z$. [1]

In alternating-current circuit analysis, the impedance of a component is modelled by a complex number (measured in ohms). Two components have impedances $Z_1=2+i$ and $Z_2=3-i$.

(a) When connected in series, the total impedance is $Z_1+Z_2$. Find this total impedance. [1]

When connected in parallel, the total impedance $Z$ satisfies $$\frac1Z=\frac1{Z_1}+\frac1{Z_2}.$$

(b) Show that $Z=\dfrac{Z_1Z_2}{Z_1+Z_2}$, and hence find $Z$ in the form $a+bi$. [3]

(c) Find $|Z|$. [2]

Let $z=a+bi$, where $a,b\in\mathbb{R}$ and $z\neq0$. On an Argand diagram, the point $P$ represents $z$ and the point $Q$ represents $iz$. $O$ is the origin.

(a) Write $iz$ in Cartesian form. [1]

(b) Show that $OP=OQ$. [2]

(c) Show that $OP$ is perpendicular to $OQ$. [2]

(d) The point $R$ is such that $OPRQ$ is a square. Given that $z=2+i$, find the complex number represented by $R$. [2]

Find all complex numbers $z$ such that $$z^2=z^*,$$ giving your answers in the form $x+iy$, where $x,y\in\mathbb{R}$.

Let $z=x+iy$, where $x,y\in\mathbb{R}$, and define $$w=\frac{z-i}{z+1},\qquad z\neq-1.$$

(a) Show that $$\operatorname{Im}(w)=\frac{y-x-1}{(x+1)^2+y^2}.$$ [4]

(b) Given that $w$ is real, find the set of points representing $z$ on an Argand diagram, describing it geometrically. [2]

(c) Given instead that $\operatorname{Re}(w)=0$, find a Cartesian equation for the set of points representing $z$, and describe this set geometrically. [3]