| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing basic complex number operations (addition, division) and simple geometric interpretation on the Argand diagram. Parts (i) and (ii) are routine calculations, part (iii) requires finding arguments and subtracting them (standard technique), and part (iv) follows directly from the parallelogram property of vector addition. All techniques are standard textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (a) State that \(u + v\) is equal to \(1 + 2i\) | B1 | [1] |
| (b) EITHER: Multiply numerator and denominator of \(u/v\) by \(3 - i\), or equivalent | M1 | |
| Simplify numerator to \(-5 + 5i\), or denominator to 10 | A1 | |
| Obtain answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 | |
| OR1: Obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | M1 | |
| Obtain \(x = -\frac{1}{2}\) or \(y = \frac{1}{2}\) | A1 | |
| Obtain answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 | |
| OR2: Using the correct processes express \(u/v\) in polar form | M1 | |
| Obtain \(x = -\frac{1}{2}\) or \(y = \frac{1}{2}\) correctly | A1 | |
| Obtain answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 | [3] |
| (ii) State that the argument of \(u/v\) is \(\frac{3}{4}\pi\) (2.36 radians or 135°) | B1√ | [1] |
| (iii) EITHER: Use facts that angle \(AOB\) = arg \(u\) – arg \(v\) and arg \(u\) – arg \(v\) = arg(\(u/v\)) | M1 | |
| Obtain given answer | A1 | |
| OR1: Obtain \(\tan\frac{AOB}\) from gradients of OA and OB and the \(\tan(A \pm B)\) formula | M1 | |
| Obtain given answer | A1 | |
| OR2: Obtain cos \(AOB\) by using the cosine formula or scalar product | M1 | |
| Obtain given answer | A1 | [2] |
| (iv) State \(OA = BC\) | B1 | |
| State \(OA\) is parallel to \(BC\) | B1 | [2] |
**(i)** (a) State that $u + v$ is equal to $1 + 2i$ | B1 | [1]
(b) EITHER: Multiply numerator and denominator of $u/v$ by $3 - i$, or equivalent | M1 |
Simplify numerator to $-5 + 5i$, or denominator to 10 | A1 |
Obtain answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 |
OR1: Obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | M1 |
Obtain $x = -\frac{1}{2}$ or $y = \frac{1}{2}$ | A1 |
Obtain answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 |
OR2: Using the correct processes express $u/v$ in polar form | M1 |
Obtain $x = -\frac{1}{2}$ or $y = \frac{1}{2}$ correctly | A1 |
Obtain answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 | [3]
**(ii)** State that the argument of $u/v$ is $\frac{3}{4}\pi$ (2.36 radians or 135°) | B1√ | [1]
**(iii)** EITHER: Use facts that angle $AOB$ = arg $u$ – arg $v$ and arg $u$ – arg $v$ = arg($u/v$) | M1 |
Obtain given answer | A1 |
OR1: Obtain $\tan\frac{AOB}$ from gradients of OA and OB and the $\tan(A \pm B)$ formula | M1 |
Obtain given answer | A1 |
OR2: Obtain cos $AOB$ by using the cosine formula or scalar product | M1 |
Obtain given answer | A1 | [2]
**(iv)** State $OA = BC$ | B1 |
State $OA$ is parallel to $BC$ | B1 | [2]7 The complex numbers $- 2 + \mathrm { i }$ and $3 + \mathrm { i }$ are denoted by $u$ and $v$ respectively.\\
(i) Find, in the form $x + \mathrm { i } y$, the complex numbers
\begin{enumerate}[label=(\alph*)]
\item $u + v$,
\item $\frac { u } { v }$, showing all your working.\\
(ii) State the argument of $\frac { u } { v }$.
In an Argand diagram with origin $O$, the points $A , B$ and $C$ represent the complex numbers $u , v$ and $u + v$ respectively.\\
(iii) Prove that angle $A O B = \frac { 3 } { 4 } \pi$.\\
(iv) State fully the geometrical relationship between the line segments $O A$ and $B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2009 Q7 [9]}}