| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| 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) is straightforward simultaneous equations with complex coefficients requiring basic algebraic manipulation. Part (b) involves standard loci (circle and half-line) and finding minimum distance, which requires geometric insight but uses routine techniques. Overall slightly easier than average due to the mechanical nature of both parts. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) EITHER: Solve for \(u\) or for \(v\) | M1 | |
| Obtain \(u=\frac{2i-6}{1-2i}\) or \(v=\frac{5}{1-2i}\), or equivalent | A1 | |
| Either: Multiply numerator and denominator by conjugate of denominator, or equivalent | M1 | |
| Or: Set \(u\) or \(v\) equal to \(x+iy\), obtain two equations by equating real and imaginary parts and solve for \(x\) or \(y\) | M1 | |
| Using \(a+ib\) and \(c+id\) for \(u\) or \(v\), equate real and imaginary parts and obtain four equations in \(a, b, c\) and \(d\) | M1 | |
| Obtain \(b+2d=2, a+2c=0, a+d=0\) and \(-b+c=3\), or equivalent | A1 | |
| Solve for one unknown | M1 | |
| Obtain final answer \(u=-2-2i\), or equivalent | A1 | |
| Obtain final answer \(v=1+2i\), or equivalent | A1 | [5] |
| (b) Show a circle with centre \(-i\) | B1 | |
| Show a circle with radius 1 | B1 | |
| Show correct half line from 2 at an angle of \(\frac{3}{4}\pi\) to the real axis | B1 | |
| Use a correct method for finding the least value of the modulus | M1 | |
| Obtain final answer \(\frac{3}{\sqrt{2}}-1\), or equivalent, e.g. 1.12 (allow 1.1) | A1 | [5] |
**(a)** **EITHER:** Solve for $u$ or for $v$ | M1 |
Obtain $u=\frac{2i-6}{1-2i}$ or $v=\frac{5}{1-2i}$, or equivalent | A1 |
Either: Multiply numerator and denominator by conjugate of denominator, or equivalent | M1 |
Or: Set $u$ or $v$ equal to $x+iy$, obtain two equations by equating real and imaginary parts and solve for $x$ or $y$ | M1 |
Using $a+ib$ and $c+id$ for $u$ or $v$, equate real and imaginary parts and obtain four equations in $a, b, c$ and $d$ | M1 |
Obtain $b+2d=2, a+2c=0, a+d=0$ and $-b+c=3$, or equivalent | A1 |
Solve for one unknown | M1 |
Obtain final answer $u=-2-2i$, or equivalent | A1 |
Obtain final answer $v=1+2i$, or equivalent | A1 | [5]
**(b)** Show a circle with centre $-i$ | B1 |
Show a circle with radius 1 | B1 |
Show correct half line from 2 at an angle of $\frac{3}{4}\pi$ to the real axis | B1 |
Use a correct method for finding the least value of the modulus | M1 |
Obtain final answer $\frac{3}{\sqrt{2}}-1$, or equivalent, e.g. 1.12 (allow 1.1) | A1 | [5]8 Throughout this question the use of a calculator is not permitted.
\begin{enumerate}[label=(\alph*)]
\item 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.
\item 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.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q8 [10]}}