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

The solutions of 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 points representing $z_1$, $z_2$, $z_1+z_2$ and $z_1-z_2$ as position vectors.

The complex number $z$ is given by $z=4-3i$.

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 correct to 2 decimal places.

Show the points representing $z$ and $z^2$ on an Argand diagram.

The complex numbers $z_1$ and $z_2$ are such that $z_2=-2+pi$ and $\frac{z_1}{z_2}=1-3i$, where $p$ is a real constant.

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 also that $\arg \left(z_1\right)={\tan }^{-1}\left(\frac{9}{7}\right)$, find the 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, in the interval $-\pi <\theta \le \pi$.

A complex number has modulus 8 and argument $\frac{\pi }{3}$.

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

$z=12-5i$

$w=3+4i$

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

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

$\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|$

$\arg \left(z_1{z}_2\right)=\arg \left(z_1\right)+\arg \left(z_2\right)$

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

$\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$

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

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

(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.

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

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

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 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 for which the equation is true,

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

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

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

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

$|z-2|=c$

(i) Draw an Argand diagram showing the locus of points representing all complex numbers $z$ for which $|z-2|=c$, and mark the points corresponding to the three roots of the equation $\text{f}\left(z\right)=0$.

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

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

Find, showing your working,

(i) $z^2$

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

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

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, 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.

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

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

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 $-\pi <\theta \le \pi$.

The solutions of the cubic equation $z^3-11z^2+36z-26=0$ are $z_1$, $z_2$ and $z_3$, where $z_3$ is real.

Given that $z_1=5+i$, work out the other two solutions of 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 $z_2$.

The complex numbers $z_1$ and $z_2$ are such that $z_1=5+pi$ and $\frac{z_1}{z_2}=-1+i$, where $p$ is a real constant.

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

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

(i) on an Argand diagram, sketch the locus 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 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.

$0\le \arg (z+2-2i)\le \frac{\pi }{4}$

$|z-3-2i|\le 5$

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 $|z+3|=c$, and mark the points corresponding to the three roots of the cubic equation.

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

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

Show all three solutions on an Argand diagram.

The complex numbers $z_1$ and $z_2$ are such that $z_1=1+pi$ and $\frac{z_1}{z_2}=-2+i$, where $p$ is a real constant.

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

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

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

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

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.

$|z-4i|>\left|z\right|$

$|z-4i|<|z+2|$

For a general complex number $z=x+iy$, where $x$ and $y$ are real and $z\ne 0$,

(i) show that $\frac{1}{z^{\star }}=\frac{z}{|z{|}^2}$,

(ii) show that $\arg \left(\frac{1}{z^{\star }}\right)=-\arg \left(z^{\star }\right)$,

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

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

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 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.

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

$|z+4-3i|\le 3$

$\frac{3\pi }{4}\le \arg (z+1-3i)\le \pi$

$|z+8|>|z+4|$

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

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 equation $\text{f}\left(z\right)=0$, showing clear algebraic working.

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

$|z-i|=c$

Draw an Argand diagram showing the locus of points representing all complex numbers $z$ for which $|z-i|=c$, and mark the points corresponding to the three roots of the equation $\text{f}\left(z\right)=0$.