Argand Diagrams (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

The solutions to the quadratic equation $z^2-8z+25=0$ are $z_1$ and $z_2$.

Work out the values of $z_1$ and $z_2$, giving your answers in the form $p\pm qi$ where $p$ and $q$ are integers.

On an Argand diagram, show the complex numbers $z_1,z_2,z_1+z_2$ and $z_1-z_2$ in position vector form.

$z=4-3i$

Show that $z^2=7-24i$

Find, showing your working:

(i) $\left|z^2\right|$

(ii) $\arg \left(z^2\right)$, giving your answer in radians to 2 decimal places.

Show $z$ and $z^2$ on an Argand diagram.

The complex numbers $z_1$ and $z_2$ are such that $z_2=-2+pi$ and $\frac{z_1}{z_2}=1-3i.$

Show that $z_1=(3p-2)+(p+6)i$.

Given that $\left|z_1\right|=\sqrt{130}$,  find the two possible values of $p$.

Given additionally that $\arg \left(z_1\right)={\tan }^{-1}\left(\frac{9}{7}\right)$,  find the precise value of $p$.

Express the complex number $-8+15i$ in the form  $r(\cos \theta +i\sin \theta )$,  where $r$ is a positive real number and $\theta$ is given in radians correct to 2 decimal places.  The value for $\theta$ should be given in the interval  $-\pi <\theta \le \pi$.

A complex number has modulus 8 and argument  $\frac{\mathrm{\pi }}{3}$.  Express the number in the form  $a+bi$, where $a$ and $b$ are real numbers.

$\begin{array}{rcl}z& =& 12-5i\\ w& =& 3+4i\end{array}$

(i) Find  $zw$, giving your answer in the form  $a+bi$  where $a$ and $b$ are real numbers.

(ii) Calculate the modulus and argument of each of the complex numbers $z,w$and $zw$, and show that these satisfy the standard results

$\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|\mathrm{and}\arg \left(z_1{z}_2\right)=\arg \left(z_1\right)+\arg \left(z_2\right).$

(i) Using your answers to part (a) and the standard results

$\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}\mathrm{and}\arg \left(\frac{z_1}{z_2}\right)=\arg \left(z_1\right)-\arg \left(z_2\right)$

       calculate the values of $\left|\frac{z}{w}\right|$ and $\arg \left(\frac{\mathrm{z}}{\mathrm{w}}\right)$.

(ii) Hence express $\frac{z}{w}$  in the form  $r(\cos \theta +i\sin \theta )$,  where $r$ is a positive real number and $\theta$ is given in radians correct to 2 decimal places.

$z_1=12\left(\cos \frac{\pi }{6}+\mathrm{i}\sin \frac{\mathrm{\pi }}{6}\right)$

$z_2=3\left(\cos \frac{5\pi }{12}+\mathrm{i}\sin \frac{5\mathrm{\pi }}{12}\right)$

Work out

(i) $z_1{z}_2$

(ii) $\frac{z_1}{z_2}$

giving your answers in modulus-argument form.

Given that  $|z-3|=|z-4i|$:

(i) On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.

(ii) Shade the region of your diagram that satisfies the inequality $|z-3|\ge |z-4i|$.

Given that $\arg (z+1)=\frac{\pi }{4}$:

(i) On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.

(ii) Shade the region of your diagram that satisfies the inequality  $0\le \arg (z+1)\le \frac{\pi }{4}$.

$f\left(z\right)=z^3-5z^2+3z-119.$

Show that $7$ and $-1+4i$ are roots of the cubic equation $f\left(z\right)=0$, and write down the third root of the equation.

Verify that there is a constant $c$ such that all three roots of the cubic equation $f\left(z\right)=0$ satisfy

$|z-2|=c$.

(i) Draw an Argand diagram showing the locus of points representing all complex numbers $z$ for which $|z-2|=c$.

Mark the points corresponding to the three roots of the cubic equation $f\left(z\right)=0$.

(ii) Shade the region of your diagram that satisfies the inequality $|z-2|\le c$.

The solutions to the cubic equation $z^3-11z^2+36z-26=0$ are $z_1,z_2$ and $z_3$, where  $z_3\in \mathrm{ℝ}$.

Given that $z_1=5+i$, work out the other two solutions to the equation.

(i) On an Argand diagram, show the complex numbers $z_1,z_2,-3z_3$ and $z_1-z_2$ as position vectors.

(ii) Describe the geometric transformation that maps $z_1$ to $z_2$.

$z=-2+\frac{15}{4}i$

Find, showing your working:

(i) $z^2$

(ii) $\left|z^2\right|$

(iii) $\arg \left(z^2\right)$, giving your answer in radians to 2 decimal places.

The complex numbers $z_1$ and $z_2$ are such that $z_1=5+pi$ and  $\frac{z_1}{z_2}=-1+i.$

Find $z_2$ in the form  $a+bi$, giving the real numbers $a$ and $b$ in terms of $p$.

Given that $\left|z_2\right|=\sqrt{73}$,  find the possible values of $p$.

Given additionally that $\arg \left(z_2\right)=2.78$  in radians to 2 decimal places, determine the exact value of  $\mathrm{Im}\left(z_2\right)$.

Express the complex number $-4-\frac{15}{2}i$  in the form  $r(\cos \theta +i\sin \theta )$,  where $r$ is a positive real number and $\theta$ is given in radians correct to 2 decimal places.  The value for $\theta$ should be given in the interval  $-\pi <\theta \le \pi$.

A complex number has modulus $\sqrt{12}$ and argument  $-\frac{5\pi }{6}$. Express the number in the form  $a+bi$, where $a$ and $b$ are real numbers.

For the general complex numbers

$z_1=r_1(\cos {\theta }_1+\mathrm{i}\sin {\theta }_1)\mathrm{and}{z}_2=r_2(\cos {\theta }_2+\mathrm{i}\sin {\theta }_2)$

given in modulus-argument form, use algebra and the appropriate trigonometric compound angle formulae to prove the results

$\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|\mathrm{and}\arg \left(z_1{z}_2\right)=\arg \left(z_1\right)+\arg \left(z_2\right).$

$z_1=9\left(\cos \frac{\pi }{6}+\mathrm{i}\sin \frac{\pi }{6}\right)$

$z_2=4\left(\cos \frac{4\mathrm{\pi }}{3}+\mathrm{i}\sin \frac{4\pi }{3}\right)$ 

Work out 

(i) $z_1{z}_2$

(ii) $\frac{z_1}{z_2}$

giving your answers in modulus-argument form with $\theta$ in the interval  $-\pi <\theta \le \pi$.

Given that $|z+1-2i|=|z-7+4i|$:

(i) On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.

(ii) Shade the region of your diagram that satisfies the inequality $|z+1-2i|>|z-7+4i|$                                     .

On an Argand diagram, sketch the loci (i.e., sets of points) for which each of the following equations is true:

(i) $\arg (z+2-2i)=\frac{\pi }{4}$

(ii) $|z-3-2i|=5$

Shade the region of your diagram that satisfies both of the following inequalities:

$0\le \arg (z+2-2i)\le \frac{\pi }{4}\mathrm{and}|z-3-2i|\le 5$

You are given that $z=-7$ satisfies the cubic equation

$z^3+9z^2+27z+91=0$

Find the other two roots of the cubic equation.

Verify that there is a constant $c$ such that all three roots of the cubic equation satisfy

$|z+3|=c$.

Draw an Argand diagram showing the locus of points representing all complex numbers $z$ for which  $|z+3|=c$.

Mark the points corresponding to the three roots of the cubic equation.

$f\left(z\right)=2z^3-13z^2+60z-100$

Given that $\frac{5}{2}$ is a root of the equation  $f\left(z\right)=0$,  use algebra to solve $f\left(z\right)=0$ completely.

Show all three solutions on an Argand diagram.

For a general complex number $z=x+iy$,  where $x,y\in \mathrm{ℝ}$ and  $z\ne 0$:

(i) Show that $\frac{1}{z^{*}}=\frac{z}{|z{|}^2}$

(ii) Show that $\arg \left(\frac{1}{\mathrm{z}*}\right)=-\arg \left(z^{*}\right).$

(iii) Given that $x<0$, write down in radians the principal argument of  $z+z^{*}$.

The complex numbers $z_1$ and $z_2$ are such that $z_1=1+pi$ and  $\frac{z_1}{z_2}=-2+i.$

Given that $\left|z_2\right|=\sqrt{58}$, find the possible values of $p$.  Be sure to show clear algebraic working.

Given additionally that $\arg \left(z_2\right)=2.09$  in radians to 2 decimal places, find the exact value of $\mathrm{Im}\left(z_2^{*}\right)$.

$z=(\sqrt{3}-3)-(\sqrt{3}+3)i$

Express $z$ in the form  $r(\cos \theta +i\sin \theta )$, where $r$ is a positive real number and $\theta$ is given as an exact value in radians.  The value for $\theta$ should be given in the interval  $-\pi <\theta \le \pi$.

A complex number $w$ is such that $\left|w\right|=\sqrt{2}\left|z\right|$ and $arg\left(w\right)=2(\pi +arg(z\left)\right)$.

Express $w$ in the form  $a+bi$, where $a$ and $b$ are real numbers.

For the general complex numbers

$z_1=r_1(\cos {\theta }_1+\mathrm{i}\sin {\theta }_1)\mathrm{and}{z}_2=r_2(\cos {\theta }_2+\mathrm{i}\sin {\theta }_2)$

given in modulus-argument form, where  $z_2\ne 0$,  use algebra and the appropriate trigonometric compound angle formulae to prove the results

$\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}\mathrm{and}\arg \left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)=\arg \left(z_1\right)-\arg \left(z_2\right).$

$z_1=-4\left(\cos \frac{\pi }{3}-\mathrm{i}\sin \frac{\mathrm{\pi }}{3}\right)$

$z_2=2\left(\cos \frac{2\pi }{3}-\mathrm{i}\sin \frac{2\pi }{3}\right)$

Re-express $z_1$ and $z_2$ in correct modulus-argument form with $\theta$ in the interval  $-\pi <\theta \le \pi$.

Work out

(i) $z_1{z}_2$

(ii) $\frac{z_2}{z_1}$

giving your answers in modulus-argument form with $\theta$ in the interval  $-\pi <\theta \le \pi$.

(i) On an Argand diagram, show the complex numbers  $z_1,z_2$ and $z_1+z_2$ as position vectors.

(ii) Use your diagram to explain briefly why we speak of a ‘parallelogram rule’ for complex number addition.

On an Argand diagram, shade the region which satisfies both of the following inequalities:

$|z-4i|>\left|z\right|\mathrm{and}|z-4i|<|z+2|$

On an Argand diagram, shade the region which satisfies all three of the following inequalities:

$$\begin{gathered} |z+4-3i|\le 3 \\ \frac{3\pi }{4}\le \arg (z+1-3i)\le \pi \\ |z+8|>|z+4| \end{gathered}$$

$f\left(z\right)=z^3+(5-7i)z^2-(36+30i)z-(150-30i)$

Given that

$z^3+(5-7i)z^2-(36+30i)z-(150-30i)=(z+(5-i\left)\right)(z^2-6iz-30)$

find all three roots of the cubic equation $f\left(z\right)=0$. Be sure to show clear algebraic working.

Verify that there is a constant $c$ such that all three roots of the cubic equation $f\left(z\right)=0$ satisfy

$|z-i|=c$

Draw an Argand diagram showing the locus of points representing all complex numbers $z$ for which $|z-i|=c$.

Mark the points corresponding to the three roots of the cubic equation  $f\left(z\right)=0$.