Complex Numbers (DP IB Applications & Interpretation (AI): HL): Exam Questions

Consider the quadratic equation $z^2-8z+25=0,z\in \mathrm{ℂ}$. 

The roots of the equation are  $z_1=a+b\mathrm{i}$  and  $z_2=a-bi$ where $a,b\in \mathrm{\mathbb{Z}}.$ 

Find the value of $a$ and $b$.

Sketch $z_1,z_2,z_1+z_2$and $z_1-z_2$ on the Argand diagram below, be sure to include an appropriate scale.

Consider the complex numbers  $z_1=-3+2\mathrm{i}$ and $z_2=1-3\mathrm{i}$.

Find

(i) $z_1+z_2$

(ii) $z_1-z_2$

(iii) $z_1{z}_2$

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

Consider the complex numbers $z_1=3-\mathrm{i}$ and  $z_2=-2-3\mathrm{i}$. 

Find the modulus and argument of $z_1{z_2}^{*}$.

Consider the complex numbers $z_1=1-2\mathrm{i}$ and $z_2=-3+5\mathrm{i}$.   

Work out the following:

(i) $\mathrm{Re}\left({\mathrm{z}}_2-{\mathrm{z}}_1\right)$

(ii) $\mathrm{Im}\left(z_1{z}_2\right)$

(iii) ${\left(\frac{z_1}{z_2}\right)}^{*}$

For part (iii) give your answer in the form $a+b\mathrm{i}$,  where $a$ and $b$ are real numbers.

Write down the complex conjugate of $z_2$ and describe the geometrical relationship between $z_2$ and ${z_2}^{*}$.

Find all possible real values for $a$ and $b$ such that

(i) $\left(a+b\mathrm{i}\right)\left(2-3\mathrm{i}\right)=8+\mathrm{i}$

(ii) $a\left(2+b\mathrm{i}\right)=b\left(-6+\mathrm{i}\right)$

(iii) $\left(2a+3\mathrm{i}\right)\left(3+b\mathrm{i}\right)=12+21\mathrm{i}$

For a general complex number $z=x+\mathrm{i}y$,  where $x,y\in \mathrm{ℝ}$,  show that

(i) $\mathrm{Re}\left(z\right)=\frac{z+z\mathit{*}}{2}$

(ii) $\mathrm{Im}\left(z\right)=\frac{z-z^{*}}{2\mathrm{i}}$

For the complex numbers $z_1=a_1+b_1\mathrm{i}$ and $z_2=a_2+b_2\mathrm{i}$,  where  $a_1,a_2,b_1,b_2\in \mathrm{ℝ}$, show that

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

Consider the complex numbers $w=2\mathrm{i}z$ and $w-z=5-5\mathrm{i}$.

Find

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

(ii) $\arg w$

(iii) $\mathrm{Re}\left(z+w\right)$

(iv) $\mathrm{Im}\left(z-w\right)$

Consider the complex numbers $z_1=a-6\mathrm{i},{\mathrm{z}}_2=1+b\mathrm{i}$ and $z_1{z}_2=-17-9\mathrm{i}$ where $a,b\in \mathrm{ℝ}$. 

Find the possible values of $a$ and $b$.

Using the answers gained in part (a), write down values for $c$ and $d$ that will satisfy the equation

$-\left(3+\mathrm{i}\right)\left(c+d\mathrm{i}\right)=17-9\mathrm{i}$

Consider the complex numbers $z=3+5\mathrm{i}\mathrm{and}w=-2+3\mathrm{i}$. 

Represent the complex numbers $z$ and $w$ on an Argand diagram.

The points $z+w$ and $z-w$ are represented by the points $\mathrm{A}$ and $\mathrm{B}$ on the Argand diagram respectively.

Find the angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}.$

Consider the complex numbers $z=-4-3\mathrm{i},w=ai$ and  $\frac{z}{w}=b+2a\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

Find the possible values of $a$ and $b$.

Find the modulus of $\frac{w}{z}$.

Let ${\omega }_1=3-\mathrm{i}\mathrm{and}{\omega }_2=1+2\mathrm{i}$.

Given that $\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}=\frac{1}{z}$, express $z$ in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

Find ${\omega }_1{\omega }_2{z}^{*}$, giving your answer in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}.$

Consider the complex number  $z=\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2}\mathrm{i}$.

Use technology to find the values of $z^2$ and $z^3$. Give your answers in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

Draw $z,z^2$ and $z^3$ on an Argand diagram.

Find the smallest integer $k>3$ such that $z^k$ is a real number.

Consider the complex numbers $z_1=\sqrt{3}+2\mathrm{i}$ and $z_2=\mathrm{i}-3\sqrt{3}$.

Find

(i) $u=z_1{z}_2$

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

The complex numbers $u$ and $v$ are represented by the points $\mathrm{A}$ and $\mathrm{B}$ respectively on an Argand diagram with origin $\mathrm{O}$. 

Determine whether the angle made by $\mathrm{OA}$ with the positive horizontal axis is greater than or less than the angle made by $\mathrm{OB}$ with the positive horizontal axis. Give a reason for your answer.

Consider the complex number $z=-a+\frac{3}{4}\mathrm{i}$.

Write down, in terms of $a$,

(i) $\mathrm{Re}\left(z^{\mathit{2}}\right)$

(ii) $\mathrm{Im}\left(z^{\mathit{3}}\right)$

In the case where $a=2$, find the modulus and argument of $z^3$.

Consider the complex numbers  $z_1=\mathrm{i}-\frac{1}{2}$ and $z_2=\frac{1}{2}-\frac{3}{\mathrm{i}}$.

Express $z_2$ in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

Find

(i) ${z_1}^{*}{z}_2$

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

(iii) $\left|\frac{z_2}{z_1}\right|$, giving your answer as an exact value.

Consider a general complex number $z=x+\mathrm{i}y$,  where  $x,y\in \mathrm{ℝ}$ , $z\in \mathrm{ℂ}$ and  $z\ne 0$. 

Show that

(i) $\mathrm{Re}\left(\frac{1}{\mathrm{z}}+\frac{1}{{\mathrm{z}}^{*}}\right)=\frac{2x}{x^{\mathit{2}}+y^{\mathit{2}}}$

(ii) $\mathrm{Im}\left(\frac{1}{z}+\frac{1}{z^{\mathit{*}}}\right)=0$

(iii) $z{z}^{*}={\left|z\right|}^2$

Consider the equation $zw-w+\mathrm{i}z+1=0$, where $w,z\in \mathrm{ℂ}$, $w=x+\mathrm{i}y$.

Find an expression in terms of $x$ and $y$ for $\mathrm{Re}\left(z\right)$.

Find in terms of $x$ given that $z$ is purely real.

Consider the complex numbers $z_1=\frac{3-\mathrm{i}}{1-2\mathrm{i}}$ and $z_2=-3\mathrm{i}+1$. 

Find the modulus of $\frac{z_1}{{z_2}^{*}}$   giving your answer as an exact value.

The argument of $\frac{z_1}{{z_2}^{*}}$ is given as $\theta ={\tan }^{-1}x$, where $0<\theta <2\mathrm{\pi }$.  Find the value of $x$.

Consider the complex numbers $z=\frac{v}{w},v=1-pi\mathrm{and}\mathrm{w}=3\mathrm{i}-2$ 

Express $z$ in the form $a+b\mathrm{i}$, where $a,b,p\in \mathrm{ℝ}.$.

In the case where $z$ is purely imaginary, represent $v,w$ and $z$ on an Argand diagram.

Consider the complex numbers $z=\frac{a-3\mathrm{i}}{2+\mathrm{i}},w=a+b\text{i}$ and $\frac{z}{w}=1+2\mathrm{i}$  where $a,b\in \mathrm{ℝ}.$

Find the values of $a$ and $b$.

Find the modulus of $\frac{w}{z}$, giving your answer as an exact value.

Find the argument of $\frac{w}{z}$ , giving your answer in the range $-\mathrm{\pi }\le \arg \frac{w}{z}\le \pi$  .

Consider the complex numbers $a-w=2z-\mathrm{i}$ and $w-2z=b\mathrm{i}-1$. 

Find the values of $a$ and $b$ such that $\mathrm{Re}\left(w\right)=\mathrm{Im}\left(z\right)$ and $\mathrm{Re}\left(w\right)=\mathrm{Re}\left(z\right)+1$.

Consider the complex numbers $z_1=5+p\mathrm{i}$, $z_2=a+b\mathrm{i}$ and $\frac{z_1}{z_2}=-1+\mathrm{i}$ , where $z\in \mathrm{ℂ}$ and $a,b\in \mathrm{ℝ}$.

Find the values of $a$ and $b$ in terms of $p$. 

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

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

Consider the complex number $z=\frac{3}{2}+\frac{\sqrt{3}}{2}\mathrm{i}$.

Use technology to find the values of $z^2$ and $z^3$. Give your answers in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

Draw $z,z^2$ and $z^3$ on an Argand diagram.

Find the smallest integer $k>3$ such that $z^k$ is purely imaginary.