| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 Part (a) requires systematic algebraic manipulation with complex conjugates (substituting w = x + iy, separating real/imaginary parts, solving simultaneous equations) - moderately challenging but routine. Part (b) requires interpreting two loci geometrically (a ray from 2i at angle π/6, and the perpendicular bisector of two points), finding their intersection algebraically, then converting to polar form - this demands strong geometric visualization, coordinate geometry, and multi-step reasoning beyond standard exercises. |
| 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(3a + 3bi + 2i(a-bi) = 17 + 8i\) | B1 | |
| Consider real and imaginary parts to obtain two linear equations in \(a\) and \(b\) | M1* | |
| Solve two simultaneous linear equations for \(a\) or \(b\) | M1 (dep*) | |
| Obtain \(7 - 2i\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either: Show or imply a triangle with side \(2\) | B1 | |
| State at least two of the angles \(\frac{1}{4}\pi\), \(\frac{2}{3}\pi\) and \(\frac{1}{12}\pi\) | B1 | |
| State or imply argument is \(\frac{1}{4}\pi\) | B1 | |
| Use sine rule or equivalent to find \(r\) | M1 | |
| Obtain \(6.69e^{\frac{1}{4}\pi i}\) | A1 | |
| Or: State \(y = x\) | B1 | |
| State \(y = \frac{1}{\sqrt{3}}x + 2\) or \(\frac{\sqrt{3}}{2} = \frac{x}{\sqrt{x^2+(y-2)^2}}\) or \(\frac{1}{2} = \frac{y-2}{\sqrt{x^2+(y-2)^2}}\) | B1 | |
| State or imply argument is \(\frac{\pi}{4}\) | B1 | |
| Solve for \(x\) or \(y\) | M1 | |
| Obtain \(6.69e^{\frac{1}{4}\pi i}\) | A1 | [5] |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $3a + 3bi + 2i(a-bi) = 17 + 8i$ | B1 | |
| Consider real and imaginary parts to obtain two linear equations in $a$ and $b$ | M1* | |
| Solve two simultaneous linear equations for $a$ or $b$ | M1 (dep*) | |
| Obtain $7 - 2i$ | A1 | [4] |
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either:** Show or imply a triangle with side $2$ | B1 | |
| State at least two of the angles $\frac{1}{4}\pi$, $\frac{2}{3}\pi$ and $\frac{1}{12}\pi$ | B1 | |
| State or imply argument is $\frac{1}{4}\pi$ | B1 | |
| Use sine rule or equivalent to find $r$ | M1 | |
| Obtain $6.69e^{\frac{1}{4}\pi i}$ | A1 | |
| **Or:** State $y = x$ | B1 | |
| State $y = \frac{1}{\sqrt{3}}x + 2$ or $\frac{\sqrt{3}}{2} = \frac{x}{\sqrt{x^2+(y-2)^2}}$ or $\frac{1}{2} = \frac{y-2}{\sqrt{x^2+(y-2)^2}}$ | B1 | |
| State or imply argument is $\frac{\pi}{4}$ | B1 | |
| Solve for $x$ or $y$ | M1 | |
| Obtain $6.69e^{\frac{1}{4}\pi i}$ | A1 | [5] |
---7
\begin{enumerate}[label=(\alph*)]
\item Without using a calculator, solve the equation
$$3 w + 2 \mathrm { i } w ^ { * } = 17 + 8 \mathrm { i }$$
where $w ^ { * }$ denotes the complex conjugate of $w$. Give your answer in the form $a + b \mathrm { i }$.
\item In an Argand diagram, the loci
$$\arg ( z - 2 \mathrm { i } ) = \frac { 1 } { 6 } \pi \quad \text { and } \quad | z - 3 | = | z - 3 \mathrm { i } |$$
intersect at the point $P$. Express the complex number represented by $P$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, giving the exact value of $\theta$ and the value of $r$ correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q7 [9]}}