| 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 using standard elimination/substitution methods. Part (b) requires sketching a circle and a half-line, then finding the minimum distance between them—this involves geometric insight but is a standard locus question. Overall slightly easier than average due to routine techniques, though part (b) requires some spatial reasoning. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply partial fractions are of the form \(\frac{A}{x-2}+\frac{Bx+C}{x^{2}+3}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A=-1, B=3, C=-1\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | [5] |
| (ii) Use correct method to obtain the first two terms of the expansions of \((x-2)^{-1}\), \(\left(1-\frac{1}{2}x\right)^{-1}\), \((x^{2}+3)^{-1}\) or \(\left(1+\frac{1}{3}x^{2}\right)^{-1}\) | M1 | |
| Substitute correct unsimplified expansions up to the term in \(x^{2}\) into each partial fraction | A1*+A1* | |
| Multiply out fully by \(Bx+C\), where \(BC\neq 0\) | M1 | |
| Obtain final answer \(\frac{1}{6}+\frac{5}{4}x+\frac{17}{72}x^{2}\), or equivalent | A1 | [5] |
**(i)** State or imply partial fractions are of the form $\frac{A}{x-2}+\frac{Bx+C}{x^{2}+3}$ | B1 |
Use a relevant method to determine a constant | M1 |
Obtain one of the values $A=-1, B=3, C=-1$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 | [5]
**(ii)** Use correct method to obtain the first two terms of the expansions of $(x-2)^{-1}$, $\left(1-\frac{1}{2}x\right)^{-1}$, $(x^{2}+3)^{-1}$ or $\left(1+\frac{1}{3}x^{2}\right)^{-1}$ | M1 |
Substitute correct unsimplified expansions up to the term in $x^{2}$ into each partial fraction | A1*+A1* |
Multiply out fully by $Bx+C$, where $BC\neq 0$ | M1 |
Obtain final answer $\frac{1}{6}+\frac{5}{4}x+\frac{17}{72}x^{2}$, or equivalent | A1 | [5]
[Symbolic binomial coefficients, e.g. $\begin{pmatrix}-1\\1\end{pmatrix}$ are not sufficient for the M1. The f.t. is on $A, B, C$.]
[In the case of an attempt to expand $\left(2x^{2}-7x-1\right)(x-2)^{-1}(x^{2}+3)^{-1}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]
[If $B$ or $C$ omitted from the form of partial fractions, give B0M1A0A0A0 in (i); M1A1*A1* in (ii)]\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 Q7 [10]}}