Argand Diagrams (Cambridge International A-Level Mathematics): Exam Questions

$z_{1} = 3 + 4 i$ and $z_{2} = 5 - 3 i$

Work out the values of $z_{1} + z_{2}$ and $z_{1} - z_{2}$.

On a sketch of an Argand diagram, show the points representing $z_{1}$, $z_{2}$, $z_{1} + z_{2}$ and $z_{1} - z_{2}$.

On a separate sketch of an Argand diagram, show $z_{1}$, $z_{2}$, $z_{1} + z_{2}$ and $z_{1} - z_{2}$ as position vectors.

Find the modulus of each of the following, giving your answers as exact values.

(i) $3 - 4 i$

(ii) $1 + i$

(iii) $- 7 i$

(iv) $- \frac{24}{25} + \frac{7}{25} i$

(v) $2$

(vi) $23 i$

(vii) $- 5 - 7 i$

(viii) $- 15$

The diagram shows how the principal argument of a complex number $z$, the argument lying in the interval $- π < \mathrm{arg} z \leq π$, is obtained from the acute angle $θ$ in each quadrant.

Write down the principal arguments of

(i) $5 i$

(ii) $3$

(iii) $- 7$

(iv) $- 4 i$

Find the principal arguments of the following complex numbers, giving your answers as exact values.

(i) $2 + 2 i$

(ii) $\sqrt{3} - i$

(iii) $- 4 + 4 \sqrt{3} i$

(iv) $- \frac{7}{2} - \frac{7}{2} i$

Find the principal arguments of the following complex numbers, giving your answers correct to 3 significant figures.

(i) $3 + 4 i$

(ii) $- 8 - 15 i$

(iii) $12 - 5 i$

(iv) $- \frac{7}{25} + \frac{24}{25} i$

The complex number $z$ is such that $z = ( 7 - p ) + ( 1 + 7 p ) i$, where $p$ is a real constant.

Show that |z| = \sqrt{50 + 50p^{2}}.

Given that |z| = 10, find the two possible values of $p$.

For each of the values of $p$ found in part (b), find the argument of the corresponding complex number $z$, giving your answers correct to 3 significant figures.

Given also that the point representing $z$ lies in the fourth quadrant of the Argand diagram, write down the value of $p$.

A complex number $z$ may be written in modulus-argument form as

$z = r ( \mathrm{cos} θ + i \mathrm{sin} θ )$

where r = |z| and $θ = \mathrm{arg} z$.

Express the complex number $- 5 + 12 i$ in modulus-argument form. Give $θ$ in radians correct to 3 significant figures, in the interval $- π < θ \leq π$.

A complex number $z$ has modulus 14 and argument $- \frac{π}{6}$.

Express $z$ in the form $a + b i$, where $a$ and $b$ are real numbers given as exact values.

Explain why each of the following is not in modulus-argument form.

(i) z_{1} = -4\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)

(ii) z_{2} = 5\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)

(iii) z_{3} = 7\left(\cos\frac{\pi}{6} + i\sin\left(-\frac{\pi}{6}\right)\right)

For two complex numbers $z_{1}$ and $z_{2}$,

|z_{1}z_{2}| = |z_{1}||z_{2}|

$\mathrm{arg} ( z_{1} z_{2} ) = \mathrm{arg} ( z_{1} ) + \mathrm{arg} ( z_{2} )$

\left|\frac{z_{1}}{z_{2}}\right| = \frac{|z_{1}|}{|z_{2}|}

\arg\left(\frac{z_{1}}{z_{2}}\right) = \arg(z_{1}) - \arg(z_{2})

Use these results to work out the following, giving your answers in modulus-argument form.

(i) 5\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) \times 4\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)

(ii) 12\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right) \div 3\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)

(iii) \sqrt{3}\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) \times \sqrt{6}\left(\cos\left(-\frac{3\pi}{4}\right) + i\sin\left(-\frac{3\pi}{4}\right)\right)

(iv) $\frac{6.25(\mathrm{cos}3.05+i\mathrm{sin}3.05)}{0.5(\mathrm{cos}1.42+i\mathrm{sin}1.42)}$

On an Argand diagram, sketch the locus of points for which

|z - (1 - 3i)| = 4

Use your sketch from part (a) to write down a complex number $z$ satisfying each of the following.

(i) |z - (1 - 3i)| < 4

(ii) |z - (1 - 3i)| > 4

On an Argand diagram, sketch the locus of points for which

|z - (3 + 5i)| = |z - (-3 - i)|

Use your sketch from part (a) to write down a complex number $z$ satisfying each of the following.

(i) |z - (3 + 5i)| < |z - (-3 - i)|

(ii) |z - (3 + 5i)| > |z - (-3 - i)|

On an Argand diagram, sketch the locus of points for which

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

Use your sketch from part (a) to write down a complex number $z$ satisfying each of the following.

(i) $0 < \mathrm{arg} ( z - ( - 2 + i ) ) < \frac{π}{4}$

(ii) $\frac{π}{4} < \mathrm{arg} ( z - ( - 2 + i ) ) < \frac{π}{2}$

The solutions of the quadratic equation $z^{2} - 8 z + 25 = 0$ are $z_{1}$ and $z_{2}$.

Work out the values of $z_{1}$ and $z_{2}$, giving your answers in the form $p \pm q i$, 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 $z = 4 - 3 i$.

Show that $z^{2} = 7 - 24 i$.

Find, showing your working,

(i) |z^{2}|

(ii) $\mathrm{arg} ( z^{2} )$, 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 $z_{2} = - 2 + p i$ and $\frac{z_{1}}{z_{2}} = 1 - 3 i$, where $p$ is a real constant.

Show that $z_{1} = ( 3 p - 2 ) + ( p + 6 ) i$.

Given that |z_{1}| = \sqrt{130}, find the two possible values of $p$.

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

Express the complex number $- 8 + 15 i$ in the form $r ( \mathrm{cos} θ + i \mathrm{sin} θ )$, where $r$ is a positive real number and $θ$ is given in radians correct to 2 decimal places, in the interval $- π < θ \leq π$.

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

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

$z = 12 - 5 i$

$w = 3 + 4 i$

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

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

|z_{1}z_{2}| = |z_{1}||z_{2}|

$\mathrm{arg} ( z_{1} z_{2} ) = \mathrm{arg} ( z_{1} ) + \mathrm{arg} ( z_{2} )$

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

\left|\frac{z_{1}}{z_{2}}\right| = \frac{|z_{1}|}{|z_{2}|}

\arg\left(\frac{z_{1}}{z_{2}}\right) = \arg(z_{1}) - \arg(z_{2})

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 $r ( \mathrm{cos} θ + i \mathrm{sin} θ )$, where $r$ is a positive real number and $θ$ 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| \geq |z - 4i|.

Given that $\mathrm{arg} ( z + 1 ) = \frac{π}{4}$,

(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 $0 \leq \mathrm{arg} ( z + 1 ) \leq \frac{π}{4}$.

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

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

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

(ii) Shade the region of your diagram that satisfies the inequality |z - 2| \leq 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 $r ( \mathrm{cos} θ + i \mathrm{sin} θ )$, where $r$ is a positive real number and $θ$ is given in radians correct to 2 decimal places, in the interval $- π < θ \leq π$.

A complex number has modulus $\sqrt{12}$ and argument $- \frac{5π}{6}$.

Express the number in the form $a + b i$, 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 $θ$ in the interval $- π < θ \leq π$.

The solutions of the cubic equation $z^{3} - 11 z^{2} + 36 z - 26 = 0$ 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 $z_{1}$, $z_{2}$, $- 3 z_{3}$ 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 $z_{1} = 5 + p i$ and $\frac{z_{1}}{z_{2}} = - 1 + i$, where $p$ is a real constant.

Find $z_{2}$ in the form $a + b i$, giving the real numbers $a$ and $b$ in terms of $p$.

Given that |z_{2}| = \sqrt{73}, find the possible values of $p$.

Given also that $\mathrm{arg} ( z_{2} ) = 2 . 78$ in radians correct to 2 decimal places, determine the exact value of $\text{Im} ( z_{2} )$.

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

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

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

$0 \leq \mathrm{arg} ( z + 2 - 2 i ) \leq \frac{π}{4}$

|z - 3 - 2i| \leq 5

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

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

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.

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

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

Show all three solutions on an Argand diagram.

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

Given that |z_{2}| = \sqrt{58}, find the possible values of $p$, showing clear algebraic working.

Given also that $\mathrm{arg} ( z_{2} ) = 2 . 09$ 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 $θ$ in the interval $- π < θ \leq π$.

Work out

(i) $z_{1} z_{2}$

(ii) $\frac{z_{2}}{z_{1}}$

giving your answers in modulus-argument form with $θ$ in the interval $- π < θ \leq π$.

(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| > |z|

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

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

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

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

(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 $r ( \mathrm{cos} θ + i \mathrm{sin} θ )$, where $r$ is a positive real number and $θ$ is given as an exact value in radians in the interval $- π < θ \leq π$.

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

Express $w$ in the form $a + b i$, 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| \leq 3

$\frac{3π}{4} \leq \mathrm{arg} ( z + 1 - 3 i ) \leq π$

|z + 8| > |z + 4|

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

Given that

$z^{3} + ( 5 - 7 i ) z^{2} - ( 36 + 30 i ) z - ( 150 - 30 i ) = ( z + ( 5 - i ) ) ( z^{2} - 6 i z - 30 )$

find all three roots of the equation $\text{f} ( z ) = 0$, showing clear algebraic working.

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