Simultaneous equations with complex numbers - Complex Numbers Arithmetic

CAIE P3 2020 June Q9

9

The complex numbers $u$ and $w$ are such that $$u - w = 2 \mathrm { i } \quad \text { and } \quad u w = 6$$ Find $u$ and $w$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
\includegraphics[max width=\textwidth, alt={}]{c1bd46f8-a33a-4927-af59-718b1c9dd4e1-16_462_709_260_719}
A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is $A$ and the radius is $r$, as shown in the diagram. The depth of water at time $t$ is $h$. At time $t = 0$ the tank is full and the depth of the water is $r$. At this instant a tap at $A$ is opened and water begins to flow out at a rate proportional to $\sqrt { h }$. The tank becomes empty at time $t = 14$. The volume of water in the tank is $V$ when the depth is $h$. It is given that $V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)$.
Show that $h$ and $t$ satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { B } { 2 r h ^ { \frac { 1 } { 2 } } - h ^ { \frac { 3 } { 2 } } } ,$$ where $B$ is a positive constant.
Solve the differential equation and obtain an expression for $t$ in terms of $h$ and $r$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P3 2020 March Q10

10

The complex numbers $v$ and $w$ satisfy the equations $$v + \mathrm { i } w = 5 \quad \text { and } \quad ( 1 + 2 \mathrm { i } ) v - w = 3 \mathrm { i } .$$ Solve the equations for $v$ and $w$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
1. On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $| z - 2 - 3 \mathrm { i } | = 1$.
2. Calculate the least value of $\arg z$ for points on this locus.
  If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

OCR Further Pure Core 2 2023 June Q2

2 In this question you must show detailed reasoning.

Write the complex number $- 24 + 7 \mathrm { i }$ in modulus-argument form.
Solve the simultaneous equations given below, giving your answers in cartesian form. $$\begin{aligned} i z + 3 w & = - 7 i
- 6 z + 5 i w & = 3 + 13 i \end{aligned}$$

CAIE P3 2013 November Q7

The complex numbers $u$ and $v$ satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for $u$ and $v$, giving both answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $| z + \mathrm { i } | = 1$ and the locus representing complex numbers $w$ satisfying $\arg ( w - 2 ) = \frac { 3 } { 4 } \pi$. Find the least value of $| z - w |$ for points on these loci.

CAIE P3 2013 November Q8

8 Throughout this question the use of a calculator is not permitted.

The complex numbers $u$ and $v$ satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for $u$ and $v$, giving both answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $| z + \mathrm { i } | = 1$ and the locus representing complex numbers $w$ satisfying $\arg ( w - 2 ) = \frac { 3 } { 4 } \pi$. Find the least value of $| z - w |$ for points on these loci.