In this question give all angr1z 0am+19tft ylweh,eo82lules in radians.

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=1−2i. ioqjdmf +kmure, w(b44-+ s/a

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 z and w correspond to the pointsfwit-a8w m,*e A and B as shown on the diagr,8fti -*amew wam below.

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 besv1it: an: icw9 .5 t:oxlv-tglow. The tangent to the circle from the point B(0,9) meets the circle at the point Ai:-15siac :xgt:lo9t .t w nvv 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 ct vwti 4+bwkby:,ri 36omplex numbers $z_1$ =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 complexlb*rk. ni x:-mhe;; ii 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. l9wp1e 0ylnp(At time t milliseconds (ms), the voltage from the fi ypl( pln09ew1rst 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 electrical circuit, the current at time t ms is tn1q 89aw u,vnn7d/iqgiven bqn8t9 7nw,da1u niqv/ y I(t)=2sin(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 seasons hotel can be modelled by the fuj: :yhqi,km.zz)c.td nction
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 hisxn ))c.y((f(hpi s nvm friends. At time t seconds, he encounters some waves with .iyx p(s(()cnm) nfvhheight
$h_1$​(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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