In this question give all angles in r5/w. lby* .ig,rhc hyiadians.

Let z=1+2i and w=4+i.

1.Find z+w=x+yi;x=  ,y=  .

2.Find:
2.1.∣z+w∣;

2.2.arg(z+w).

3.Find θ, the angle shown on the diagram below.

  Let z=2+i and w=1−2il n-5a*bueaxs:m02mo .

1.Find zw=x-yi ; x=  ,y=  .

2.Illustrate z , w and zw on the same Argand diagram.

3.let θ be the angle between zw and w. Find θ, giving your answer in radians.
Hence we get θ(≈)  .

  The complex numbers z1 :3p b5mptrzy xd5 ,r3cc4fth and w correspond to the points A and B as shown on the diagram below. tm3zfrd by,34t:5hxrcp 15 cp

1.Find the exact value of ∣z−w∣.
∣z−w∣=x$\sqrt{y-\sqrt{z}}$; x=  , y=  , z=  .
2.1.Find the exact perimeter of triangle AOB.
perimeter=x+∣z−w∣; x=  .
2.2.Find the exact area of triangle AOB.
area=$\frac{x}{y}$ ; x=  , y=  .

  A circle is drawn on an Argand diagram as shown below. The tangent to the circlecvi 2h y4ekf:7u-o3o g from the point B(0,9) meets the circle at the o2 o-g:3ue7 cifhkyv4point A as shown. Let w= OA$^{→}$.

1.Show that ∣w∣=3$\sqrt{3}$.

2.Find arg w.w=$\frac{\pi}{x}$,x=  .

3.Hence write w in the form a+bi where a,b∈R.
w=$\frac{3\sqrt{3}}{x}+\frac{y}{2}i$,x=  , y=  .

  Let $z_1$ =2$\sqrt{3} cis( $\frac{7π}{12}$), $z_3$=2cisθ, 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 θ= $\frac{π}{12}$.

2.Find arg($z_1$-$z_2$)=-$\frac{xπ}{y}$, x=  ,y=  .

3.Express $z_2$ in modulus-argument form.$z_2$=xcis($\frac{yΠ}{z}$),x=  ,y=  ,z=  .

On an Argand diagram, the complex nu1inek)/; rnmtyz (ba1mbers $z_1$ =2+2$\sqrt{3}$i, $z_2$=1−i and $z_3$=$z_1$$z_1$ are represented by the vertices of a triangle.

Find the area of the triangle.

  Points A and B represent the cowj/t3ah8bn2s 4a4 kj b 1,sybgmplex numbers $z_1$ = 3​ −i and $z_2$=-3-3i as shown on an Argand diagram below.

1.Find the angle AOB=$\frac{x\pi}{y}$ ; x=  ,y=  .

2.Find the argument of $z_1$$z_2$
arg$z_1$$z_2$=-$frac{x\pi}{y}$ ; x=  ,y=  .

3.Given that the real powers of p$z_1$$z_2$ , for p>0, all lie on a unit circle centred at the origin, find the exact value of p.
p=$\frac{\sqrt{x}}{y}$ ; x=  ,y=  .

  Let z=r$e^{i\frac{\pi}{3}}$ where r∈R$^+$.
1.For r= $sqrt{2}$,

1.1.express $z^2$ and $z^3$ in the form a+bi where a,b∈R;
$z^2$=-1+$\sqrt{x}$i and $z^3$=-2$\sqrt{2}$+yi; x=  ,y=  .

1.2.draw $z^2$ and $z^3$ on the following Argand diagram.

2.Given that the integer powers of w=(3−3i)z lie on a unit circle centred at the origin, find the value of r.
r=$\frac{\sqrt{x}}{y}$ ; x=  .y=  .

  Let z=r$e^{i\frac{\pi}{3}}$ where r∈R$^+$.
1.For r= $sqrt{3}$,

1.1.express $z^2$ and $z^3$ in the form a+bi where a,b∈R;
$z^2$=$\frac{3}{2}$+$\frac{3\sqrt{x}}{y}$i and $z^3$=0+$3sqrt{z}$i; x=  ,y=  ,z=  .

1.2.draw $z^2$ and $z^3$ on the following Argand diagram.

2.Given that the integer powers of w=$\frac{z}{6+2i}$ lie on a unit circle centred at the origin, find the value of r.
r=$x\sqrt{y}$ ; x=  .y=  .

  Let z=$\sqrt{2}cis(\frac{3\pi}{8})$ and w=2cis($\frac{n\pi}{24}$)，where n∈$Z^+$.

1.Find the value of $z^6$ . Give your answer in the form re$^{iθ}$, where r≥0, -Π＜θ≤Π.
$z^6$=xe$^{i\frac{Π}{y}}$ . x=  ,y=  .
2.Find the value of $(wz)^4$ for n=5 . Give your answer in the form re$^{iθ}$, where r≥0, -Π＜θ≤Π.
$(wz)^4$=xe$^{i\frac{Π}{y}}$ . x=  ,y=  .
3.Find the smallest integer n>9 such that $\frac{z}{w}$ ∈R.
n=  .

  Two voltage sources are connected to a circuit. At time t millisecons0si4x t2jh8vdns u6w a.:fp/ds (ms), the v.hw8v2d4j a /sxuf :t 6ps0isnoltage from the first source is $V_1$(t)=12cos(20t) and the voltage from the second source is $V_2$(t)=18cos(20t+5), shere both $V_1$(t) and $V_2$(t) are
easured in volts.
1.Write, in the form V(t)=Acos(ωt+φ), an expression for the total voltage in the circuit at time t ms.
2.Hence write down the highest voltage in the circuit.
Hence the highest voltage in the circuit is    volts
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  In an unbalanced three-phase el17,mzud0bqk oectrical circuit, the current at time t ms is given by I(t)=2sinuko7qz,d 0b1m (5t)+5sin(5t-$\frac{5\pi}{4}$),
where I(t) is measured in milliamperes (mA).

1.Write I(t) in the form Acos(ωt+φ).

2.Hence find the highest current flowing through the circuit, and the time it first occurs.
Hence the highest current is   mA and the 1st time it occurs is at   ms

  The revenues of a four sease ovzt-hxf0; a2b8gk k y2s1c9ons hotel can be modelled by the function
R(t)=58.2sin(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
C(t)=31.4sin(0.0172t+1.14)+85.0.

Both R(t) and C(t) are measured in thousand dollars.

1.Show that the profits of the hotel can be modelled by the function P(t)=83.9sin(0.0172t−1.51)+119.

2.According to the model, find:

2.1.the highest profit the hotel will make;
According to the model, the highest profit is
$P_{max}$≈  thousand.

2.2.the date on which the highest profit will occur.
Hence the highest profit occurs on day    or June   .
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  Ali is swimming in a public pool with some of his friends. At t6.k*qxuw i2:syr. p8tle6qeqqfy q2 0ime t seconds, he encounters some waves wiqp.si66qyl*f 2q.:eq kx0yt qr28uewth height
$h_1$​(t)=0.15sin(3t) from big Bobby jumping into the pool, and waves of height
$h_2$(t)=0.08sin(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)=Asin(ωt+φ), an expression for the total height of the waves Ali encounters at time t seconds.

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

3.Find the first time when the waves reaching Ali has maximum height.
Hence the first time when the wave reaching Ali has max height is at   s
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