| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 Part (a) is a routine algebraic manipulation to find a complex number from its conjugate equation. Part (b)(i) requires sketching a standard circle with radius 1 and a horizontal line, then shading their intersection—straightforward geometric interpretation. Part (b)(ii) involves finding arguments at boundary points of the region, requiring some geometric visualization but using standard techniques. This is a typical multi-part question slightly easier than average due to the straightforward nature of each component. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Substitute and obtain a correct equation in \(x\) and \(y\) | B1 | |
| Use \(i^2 = -1\) and equate real and imaginary parts | M1 | |
| Obtain two correct equations, e.g. \(x + 2y + 1 = 0\) and \(y + 2x = 0\) | A1 | |
| Solve for \(x\) or for \(y\) | M1 | |
| Obtain answer \(z = \frac{1}{5} - \frac{2}{5}i\) | A1 | [5] |
| (b) (i) Show a circle with centre \(-1 + 3i\) | B1 | |
| Show a circle with radius 1 | B1 | |
| Show the line \(\text{Im}z = 3\) | B1 | |
| Shade the correct region | B1 | [4] |
| (ii) Carry out a complete method to calculate the relevant angle | M1 | |
| Obtain answer 0.588 radians (accept 33.7°) | A1 | [2] |
(a) Substitute and obtain a correct equation in $x$ and $y$ | B1 |
Use $i^2 = -1$ and equate real and imaginary parts | M1 |
Obtain two correct equations, e.g. $x + 2y + 1 = 0$ and $y + 2x = 0$ | A1 |
Solve for $x$ or for $y$ | M1 |
Obtain answer $z = \frac{1}{5} - \frac{2}{5}i$ | A1 | [5]
(b) (i) Show a circle with centre $-1 + 3i$ | B1 |
Show a circle with radius 1 | B1 |
Show the line $\text{Im}z = 3$ | B1 |
Shade the correct region | B1 | [4]
(ii) Carry out a complete method to calculate the relevant angle | M1 |
Obtain answer 0.588 radians (accept 33.7°) | A1 | [2]10
\begin{enumerate}[label=(\alph*)]
\item Find the complex number $z$ satisfying the equation $z ^ { * } + 1 = 2 \mathrm { i } z$, where $z ^ { * }$ denotes the complex conjugate of $z$. Give your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $| z + 1 - 3 \mathrm { i } | \leqslant 1$ and $\operatorname { Im } z \geqslant 3$, where $\operatorname { Im } z$ denotes the imaginary part of $z$.
\item Determine the difference between the greatest and least values of $\arg z$ for points lying in this region.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2016 Q10 [11]}}