| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This question involves standard complex number techniques: (a)(i) requires routine multiplication by conjugate to express in Cartesian form, (a)(ii) uses basic argument properties, and (b) requires sketching standard loci (circle and perpendicular bisector). All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks |
|---|---|
| Multiply numerator and denominator by \(a - 2i\), or equivalent | M1 |
| Obtain final answer \(\frac{5a}{a^2 + 4} - \frac{10i}{a^2 + 4}\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain two equations in \(x\) and \(y\), solve for \(x\) or for \(y\) | M1 | |
| Obtain final answer \(x = \frac{5a}{a^2 + 4}\) and \(y = \frac{10}{a^2 + 4}\), or equivalent | A1 | [2] |
| (ii) Either state \(\arg(u) = -\frac{3}{4}\pi\), or express \(u^*\) in terms of \(a\) (f.t. on \(u\)) | B1√ | |
| Use correct method to form an equation in \(a\), e.g. \(5a = -10\) | M1 | |
| Obtain \(a = -2\) correctly | A1 | [3] |
| (b) Show a point representing \(2 + 2i\) in relatively correct position in an Argand diagram | B1 | |
| Show the circle with centre at the origin and radius 2 | B1 | |
| Show the perpendicular bisector of the line segment from the origin to the point representing \(2 + 2i\) | B1√ | |
| Shade the correct region | B1 | [4] |
**(a)(i) EITHER:**
Multiply numerator and denominator by $a - 2i$, or equivalent | M1 |
Obtain final answer $\frac{5a}{a^2 + 4} - \frac{10i}{a^2 + 4}$, or equivalent | A1 |
**OR:**
Obtain two equations in $x$ and $y$, solve for $x$ or for $y$ | M1 |
Obtain final answer $x = \frac{5a}{a^2 + 4}$ and $y = \frac{10}{a^2 + 4}$, or equivalent | A1 | [2]
**(ii)** Either state $\arg(u) = -\frac{3}{4}\pi$, or express $u^*$ in terms of $a$ (f.t. on $u$) | B1√ |
Use correct method to form an equation in $a$, e.g. $5a = -10$ | M1 |
Obtain $a = -2$ correctly | A1 | [3]
**(b)** Show a point representing $2 + 2i$ in relatively correct position in an Argand diagram | B1 |
Show the circle with centre at the origin and radius 2 | B1 |
Show the perpendicular bisector of the line segment from the origin to the point representing $2 + 2i$ | B1√ |
Shade the correct region | B1 | [4]
[SR: Give the first B1 and the B1√ for obtaining $y = 2 - x$, or equivalent, and sketching the attempt.]7
\begin{enumerate}[label=(\alph*)]
\item The complex number $u$ is defined by $u = \frac { 5 } { a + 2 \mathrm { i } }$, where the constant $a$ is real.
\begin{enumerate}[label=(\roman*)]
\item Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item Find the value of $a$ for which $\arg \left( u ^ { * } \right) = \frac { 3 } { 4 } \pi$, where $u ^ { * }$ denotes the complex conjugate of $u$.
\end{enumerate}\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ which satisfy both the inequalities $| z | < 2$ and $| z | < | z - 2 - 2 \mathrm { i } |$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2011 Q7 [9]}}