| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Simultaneous equations with complex numbers |
| Difficulty | Standard +0.3 Part (a) involves solving simultaneous equations with complex numbers using substitution and the quadratic formula—a standard technique. Part (b) requires sketching a region on an Argand diagram satisfying three inequalities, which is routine practice for P3/Further Pure. Both parts are straightforward applications of well-practiced methods with no novel insight required. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Eliminate \(u\) or \(w\) and obtain an equation in \(w\) or \(u\) | M1 | |
| Obtain a quadratic in \(u\) or \(w\), e.g. \(u^2-2iu-6=0\) or \(w^2+2iw-6=0\) | A1 | |
| Solve a 3-term quadratic for \(u\) or for \(w\) | M1 | |
| Obtain answer \(u=\sqrt{5}+\mathrm{i}\), \(w=\sqrt{5}-\mathrm{i}\) | A1 | |
| Obtain answer \(u=-\sqrt{5}+\mathrm{i}\), \(w=-\sqrt{5}-\mathrm{i}\) | A1 | |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show the point representing \(2+2\mathrm{i}\) | B1 | |
| Show a circle with centre \(2+2\mathrm{i}\) and radius \(2\) | B1 FT | FT is on the position of \(2+2\mathrm{i}\) |
| Show half-line from origin at \(45°\) to the positive \(x\)-axis | B1 | |
| Show line for \(\text{Re } z=3\) | B1 | |
| Shade the correct region | B1 | |
| 5 |
## Question 9:
### Part 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Eliminate $u$ or $w$ and obtain an equation in $w$ or $u$ | M1 | |
| Obtain a quadratic in $u$ or $w$, e.g. $u^2-2iu-6=0$ or $w^2+2iw-6=0$ | A1 | |
| Solve a 3-term quadratic for $u$ or for $w$ | M1 | |
| Obtain answer $u=\sqrt{5}+\mathrm{i}$, $w=\sqrt{5}-\mathrm{i}$ | A1 | |
| Obtain answer $u=-\sqrt{5}+\mathrm{i}$, $w=-\sqrt{5}-\mathrm{i}$ | A1 | |
| | **5** | |
### Part 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show the point representing $2+2\mathrm{i}$ | B1 | |
| Show a circle with centre $2+2\mathrm{i}$ and radius $2$ | B1 FT | FT is on the position of $2+2\mathrm{i}$ |
| Show half-line from origin at $45°$ to the positive $x$-axis | B1 | |
| Show line for $\text{Re } z=3$ | B1 | |
| Shade the correct region | B1 | |
| | **5** | |
---9
\begin{enumerate}[label=(\alph*)]
\item 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.
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities
$$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c1bd46f8-a33a-4927-af59-718b1c9dd4e1-16_462_709_260_719}
\end{center}
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 $V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)$.\\
(a) 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.\\
(b) 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.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q9 [10]}}