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Question 1

HLPaper 1

Let $A(t)=5.1 \sin (\omega t)$ and $B(t)=3.2 \sin (\omega t+5)$

1.

Find $A(t)+B(t)$ in the form $a \sin (\omega t+B)$ .

[4]

Question 2

HLPaper 1

[Maximum Mark : 7] Let $z_{1}=a\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$ and $z_{2}=b\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$ Express $\left(\frac{z_{2}}{z_{1}}\right)^{2}$ in Cartesian form.

1.

[Maximum Mark : 7] Let $z_{1}=a\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$ and $z_{2}=b\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$ Express $\left(\frac{z_{2}}{z_{1}}\right)^{2}$ in Cartesian form.

[3]

Question 3

HLPaper 1

Let $A(t)=10 \cos (3 t+4)$ and $B(t)=20 \cos (3 t+5)$ $A(t)$ and $B(t)$ are the real parts of the complex numbers $z_{A}$ and $z_{B}$ respectively.

1.

Express $z_{A}$ and $z_{B}$ in the form $r \operatorname{cis}(\theta)$ and $r e^{i \theta}$ , where $\theta$ is in terms of $t$ .

[3]

2.

Find $z_{A}+z_{B}$ in the form $r e^{i \theta}$ , where $\theta$ is in terms of $t$ .

[3]

Question 4

HLPaper 1

Let $z_{1}=\frac{\sqrt{6}-i \sqrt{2}}{2}$ and $z_{2}=1-i$

1.

Find the value of $\frac{z_{1}}{z_{2}}$ in the form of $a+b i$

[2]

2.

Write $z_{1}$ and $z_{2}$ in the form $r \operatorname{cis}(\theta)$ where $r>0$

[2]

3.

Show that $\frac{z_{1}}{z_{2}}=\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}$

[4]

4.

Find the exact values of $\cos \frac{\pi}{12}$ and $\sin \frac{\pi}{12}$ . Justify your answer.

[2]

Question 5

HLPaper 2

In a physics simulation, two forces are represented as complex numbers: Let z = r cis θ and w = 3 cis 5

1.

Express z/w in terms of r and θ. Let z = r cis θ and w = 3 cis 5

[2]

2.

Express w/z in terms of r and θ. Let z = r cis θ and w = 3 cis 5

[2]

3.

Find z/w² in terms of r and θ. Let z = r cis θ and w = 3 cis 5

[2]

Question 6

HLPaper 2

In a telecommunications project, engineers use complex numbers to model signal processing.

1.

Find the polar form for $z = 1 + i√3$

[3]

2.

Find the polar form for $w = 2 − i2$

[2]

3.

Find the polar form of $z⋅w$

[2]

4.

Find the polar form of $w/z$

[2]

Question 7

HLPaper 1

Consider $w = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$

These four points form the vertices of a quadrilateral, Q.

1.

Express $w^2$ and $w^3$ in modulus-argument form.

[3]

2.

Sketch on an Argand diagram the points represented by $w^0$ , $w^1$ , $w^2$ and $w^3$ .

[2]

3.

Show that the area of the quadrilateral Q is $\frac{21\sqrt{3}}{2}$ .

[3]

4.

Let $z = 2(\cos\frac{\pi}{n} + i\sin\frac{\pi}{n})$ , $n \in \mathbb{Z}^+$ . The points represented on an Argand diagram by $z^0$ , $z^1$ , $z^2$ , $\ldots$ , $z^n$ form the vertices of a polygon $P_n$ .

Show that the area of the polygon $P_n$ can be expressed in the form $a(b^n - 1)\sin\frac{\pi}{n}$ , where $a, b \in \mathbb{R}$ .

[6]

Question 8

HLPaper 1

Consider the complex numbers ${z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}$ and $w = \frac{{{z_1}}}{{{z_2}}}$ .

1.

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write downthe modulus of $w$ ;

2.

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write downthe argument of $w$ .

3.

Find the smallest positive integer value of $n$ , such that ${w^n}$ is a real number.

Question 9

HLPaper 2

In a digital art project, artists use complex numbers to create intricate designs. Each design element is represented in Cartesian form, but they need to convert them to polar form for rendering.

1.

Find the polar form r cis θ of z₁ = 1 + i

[2]

2.

Find the polar form r cis θ of z₂ = −1 + i

[2]

3.

Find the polar form r cis θ of z₃ = −1 − i

[2]

4.

Find the polar form r cis θ of z₄ = 1 − i

[2]

Question 10

HLPaper 1

Let $z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi$ .

1.

Solve $2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ$ .

[5]

2.

Show that $\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}$ .

[3]

3.

Find the modulus and argument of $z$ in terms of $\theta$ . Express each answer in its simplest form.

[9]

4.

Hence find the cube roots of $z$ in modulus-argument form.

[5]

AHL 1.13—Complex numbers continued Questionbank