Complex Numbers

[Maximum mark: 5]

In this question give all angles in radians.

Let $z = 1 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w = 4 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Find $z+w$. [1]
   

2. Find:

   1. $|z+w|$;

   2. $\arg(z+w)$. [2]
      

3. Find $\theta$, the angle shown on the diagram below. [2]
   

   

[Maximum mark: 7]

Let $z = 2+{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w = 1-2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

1. Find $zw$. [2]
   

2. Illustrate $z$, $w$ and $zw$ on the same Argand diagram. [3]
   

3. Let $\theta$ be the angle between $zw$ and $w$. Find $\theta$, giving your answer in radians.[2]
   

[Maximum mark: 6]

The complex numbers $z$ and $w$ correspond to the points A and B as shown on the diagram below.

1. Find the exact value of $|z - w|$. [2]
   

2. 1. Find the exact perimeter of triangle AOB.

   2. Find the exact area of triangle AOB. [4]
      

[Maximum mark: 6]

Let $z_1 = 2\sqrt{3}\mathop{\mathrm{cis}}\hspace{-0.1em}\Big(\dfrac{7\pi}{12}\Big)$, $z_3 = 2\mathop{\mathrm{cis}}\theta$, and $z_2 = z_1 + z_3$ be represented by the points

A, B and C on an Argand diagram as shown below.

The shape OABC is a rectangle.

1. Show that $\theta = \dfrac{\pi}{12}$. [1]
   

2. Find $\arg\hspace{0.05em}(z_1 - z_2)$. [1]
   

3. Express $z_2$ in modulus-argument form. [4]
   

[Maximum mark: 6]

A circle is drawn on an Argand diagram as shown below. The tangent to the circle from the point B$(0,9)$ meets the circle at the point A as shown. Let $w = \vv{\mathrm{OA}}$.

1. Show that $|w| = 3\sqrt{3}$. [2]
   

2. Find $\arg w$. [2]
   

3. Hence write $w$ in the form $a+b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [2]
   

[Maximum mark: 6]

On an Argand diagram, the complex numbers $z_1 = 2 + 2\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$, $z_2 = 1 - {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_3 = z_1z_2$ are represented by the vertices of a triangle.

Find the area of the triangle.

[Maximum mark: 8]

Consider two power sources with voltages $V_1$ and $V_2$ respectively, where $\omega$ is the frequency of the power source in Hertz (Hz) and $t$ is time in seconds.

It is given that

When the two power sources are combined, the resultant voltage $V_1 + V_2$ is given by

1. Find $V_2$. Give your answer in the form $a\sin(\omega t + b)$, where $a,b \in \mathbb{R}$.[6]
   

2. Hence, determine the difference in phase shift between $V_1$ and $V_2$.[2]
   

[Maximum mark: 6]

Points A and B represent the complex numbers $z_1 = \sqrt{3} - {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_2 = -3 - 3{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ as shown on an Argand diagram below.

1. Find the angle AOB. [2]
   

2. Find the argument of $z_1z_2$. [1]
   

3. Given that the real powers of $pz_1z_2$, for $p > 0$, all lie on a unit circle centred at the origin, find the exact value of $p$. [3]
   

[Maximum mark: 8]

Let $z = \sqrt{2}\mathop{\mathrm{cis}}\hspace{-0.1em}\left(\dfrac{3\pi}{8}\right)$ and $w = 2\mathop{\mathrm{cis}}\hspace{-0.1em}\bigg(\dfrac{n\pi}{24}\bigg)$ , where $n \in \mathbb{Z}^+$.

1. Find the value of $z^6$. Give your answer in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r \geq 0$, $-\pi < \theta \leq \pi$. [2]
   

2. Find the value of $(wz)^4$ for $n = 5$. Give your answer in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r \geq 0$, $-\pi < \theta \leq \pi$. [3]
   

3. Find the smallest integer $n > 9$ such that $\dfrac{z}{w} \in \mathbb{R}$. [3]
   

[Maximum mark: 6]

Let $z = re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{\pi}{3}}$ where $r \in \mathbb{R}^+$.

1. For $r = \sqrt{2}$,

   1. express $z^2$ and $z^3$ in the form $a+b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$;

   2. draw $z^2$ and $z^3$ on the following Argand diagram. [4]
      

      

2. Given that the integer powers of $w = (3-3{\mathrm{\hspace{0.05em}i}\mkern 1mu})\hspace{0.05em}z$ lie on a unit circle centred at the origin, find the value of $r$. [2]
   

[Maximum mark: 6]

Let $z = re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\frac{\pi}{6}}$ where $r \in \mathbb{R}^+$.

1. For $r = \sqrt{3}$,

   1. express $z^2$ and $z^3$ in the form $a+b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$;

   2. draw $z^2$ and $z^3$ on the following Argand diagram. [4]
      

      

2. Given that the integer powers of $w = \dfrac{z}{6+2{\mathrm{\hspace{0.05em}i}\mkern 1mu}}$ lie on a unit circle centred

   at the origin, find the value of $r$. [2]
   

[Maximum mark: 5]

Two voltage sources are connected to a circuit. At time $t$ milliseconds (ms), the voltage from the first source is $V_1(t) = 12\cos(20t)$ and the voltage from the second source is $V_2(t) = 18\cos(20t+5)$, where both $V_1(t)$ and $V_2(t)$ are measured in volts.

1. Write, in the form $V(t) = A\cos\hspace{0.05em}(\omega t+\varphi)$, an expression for the total voltage in the circuit at time $t$ ms. [4]
   

2. Hence write down the highest voltage in the circuit. [1]
   

[Maximum mark: 6]

Ali is swimming in a public pool with some of his friends. At time $t$ seconds, he $\text{encounters}$ some waves with height $h_1(t) = 0.15\sin\hspace{0.05em}(3t)$ from big Bobby jumping into the pool, and waves of height $h_2(t) = 0.08\sin\hspace{0.05em}(3t+1.25)$ from small Suzie jumping into the pool. Both $h_1(t)$ and $h_2(t)$ are measured in metres.

1. Write, in the form $h(t) = A\sin\hspace{0.05em}(\omega t+\varphi)$, an expression for the total height of the waves Ali encounters at time $t$ seconds. [3]
   

2. Find the times in the first $5$ seconds when Ali isn't affected by any waves. [2]
   

3. Find the first time when the waves reaching Ali has maximum height. [1]
   

[Maximum mark: 6]

The revenues of a four seasons hotel can be modelled by the function

$\mathrm{R}(t) = 58.2\sin\hspace{0.05em}(0.0172t - 1.25) + 204$,

where $t$ is the number of days after midnight on $31$ December.

In a similar way, the operating costs of the hotel can be modelled by the function

$\mathrm{C}(t) = 31.4\sin\hspace{0.05em}(0.0172t + 1.14) + 85.0$.

Both $\mathrm{R}(t)$ and $\mathrm{C}(t)$ are measured in thousand dollars.

1. Show that the profits of the hotel can be modelled by the function $\mathrm{P}(t) = 83.9\sin\hspace{0.05em}(0.0172t-1.51) + 119$. [3]
   

2. According to the model, find:

   1. the highest profit the hotel will make;

   2. the date on which the highest profit will occur. [3]
      

[Maximum mark: 6]

In an unbalanced three-phase electrical circuit, the current at time $t$ ms is given by

$I(t) = 2\sin\hspace{0.05em}(5t) + 5\sin\hspace{-0.1em}\Big(5t-\dfrac{3\pi}{4}\Big) + 10\sin\hspace{-0.1em}\Big(5t-\dfrac{5\pi}{4}\Big)$,

where $I(t)$ is measured in milliamperes (mA).

1. Write $I(t)$ in the form $A\cos\hspace{0.05em}(\omega t+\varphi)$. [4]
   

2. Hence find the highest current flowing through the circuit, and the time it first occurs. [2]