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

Solve each equation for $x\in \mathrm{ℂ}$.

(i) $x^2+4x+5=0$

(ii) $x^2=-625$

(iii) $x^4=24-2x^2$

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

(i) $a+b\mathrm{i}=8\mathrm{i}$

(ii) $\left(2+3\mathrm{i}\right)\left(a+b\mathrm{i}\right)=13$

(iii) $\left(a+\mathrm{i}\right)\left(2+b\mathrm{i}\right)=-6+22\mathrm{i}$.

The complex numbers $z$ and $w$ satisfy $w=\mathrm{i}z$ and $w+2z=7+6\mathrm{i}$.

Find

(i) $\text{Re}\left(w\right)$

(ii) $\text{Im}\left(w\right)$

(iii) $\text{Re}\left(z\right)$

(iv) $\text{Im}\left(z\right)$.

Find the complex numbers $z$ and $w$ such that 

$2z-\mathrm{i}{w}^{*}=5+7\mathrm{i}$ 

$w+\mathrm{i}{z}^{*}=5+16\mathrm{i}$

Let $z=3+8\mathrm{i}$ and $w=4-4\mathrm{i}$. The diagram below shows $z$, $w$ and $z+w$ on an Argand diagram.

Find $\theta$, the angle shown on the diagram.

Find the area of the triangle shown in the diagram.

Let $z=-1-3\mathrm{i}$ and $w=1+\mathrm{i}$.

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

Plot $z$, $w$ and $zw$ on the Argand diagram below.

Let $\theta$ be the angle between $z$ and $zw$, and let $\varphi$ be the angle between $w$ and $zw$, both measured at the origin.

Find $\theta$ and $\varphi$, giving your answers in degrees.

Consider the equation $x^2+bx+c=0.$

Write down an inequality, in terms of $b$ and $c,$ that shows the equation has no real solutions.

$5—3i$is one solution to the equation $x^2+bx+c=0.$

Find the values of $b$ and $c.$

Let $z=c+bi.$

Find $z^5$ using technology.

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

Plot $z$ and $w$ on an Argand diagram.

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

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

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

The roots of the equation are $z_1=a+b\mathrm{i}$ and $z_2=a-b\mathrm{i}$, where $a,b\in \mathrm{\mathbb{Z}}$ and $b>0$.

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

Plot $z_1$, $z_2$, $z_1+z_2$ and $z_1-z_2$ on an Argand diagram.

Let ${\omega }_1=3-\mathrm{i}$ 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 $a,b\in \mathrm{ℝ}$.

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

Find

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

(ii) $\text{arg}\left(w\right)$

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

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

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

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

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

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

Find the modulus and the argument of $z_1{\left(z_2\right)}^{*}$.

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

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

The argument of $\frac{z_1}{{\left(z_2\right)}^{*}}$ is $\theta$, where $0<\theta <2\pi$.

Find the exact value of $\tan \theta$.

Consider the complex numbers $v=1-p\mathrm{i}$ and $w=3\mathrm{i}-2$, where $p\in \mathrm{ℝ}$, and let $z=\frac{v}{w}$.

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

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

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 the complex number $z=-a+\frac{3}{4}\mathrm{i}$, where $a\in \mathrm{ℝ}$.

Find, in terms of $a$,

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

(ii) $\text{Im}\left(z^3\right)$.

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

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

Show that

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

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

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

Find all possible real values of $a$ 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 $z=x+y\mathrm{i}$, where $x,y\in \mathrm{ℝ}$, show that

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

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

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 $z_1=a-6\mathrm{i}$ and $z_2=1+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

Given that $z_1{z}_2=-17-9\mathrm{i}$, find the possible values of $a$ and $b$.

Using your answers to part (a), write down values of $c$ and $d$ that satisfy the equation

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

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.

Let $w=\frac{z+1}{z^{*}+1}$, where $z=a+b\mathrm{i}$ and $a,b\in \mathrm{ℝ}$.

Write $w$ in the form $x+y\mathrm{i}$, where $x,y\in \mathrm{ℝ}$.

Determine the conditions on $a$ and $b$ for which $w$ is purely imaginary.

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

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 $-\pi <\text{arg}\left(\frac{w}{z}\right)\le \pi$.

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

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

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) $v=\frac{z_1}{z_2}$.

Determine whether $\text{arg}\left(u\right)$ is greater than or less than $\text{arg}\left(v\right)$, where $-\pi <\text{arg}\left(u\right)\le \pi$ and $-\pi <\text{arg}\left(v\right)\le \pi$. Give a reason for your answer.

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

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

Given that $z$ is purely real, find $z$ in terms of $x$.

The complex numbers $z$ and $w$ satisfy $a-w=2z-\mathrm{i}$ and $w-2z=b\mathrm{i}-1$, where $a,b\in \mathrm{ℝ}$.

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

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.