| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| 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 standard textbook exercise on finding square roots of complex numbers using the algebraic method (equating real and imaginary parts), requiring routine manipulation but no insight. Part (b) involves sketching standard loci (a filled circle and an angular sector), which is mechanical application of definitions. Both parts are slightly easier than average A-level questions due to their routine nature and clear structure. |
| Spec | 4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | Square x + iy and equate real and imaginary parts to 8 and –15 | M1 |
| Obtainx2 − y2 =8and 2xy=−15 | A1 | |
| Eliminate one unknown and find a horizontal equation in the other | M1 | |
| Obtain 4x4 −32x2 −225=0or4y4 +32y2 −225=0, or three term equivalent | A1 |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(b) | Show a circle with centre2+i in a relatively correct position | B1 |
| Show a circle with radius 2 and centre not at the origin | B1 |
| Answer | Marks |
|---|---|
| 4 | B1 |
| Shade the correct region | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(a) ---
7(a) | Square x + iy and equate real and imaginary parts to 8 and –15 | M1
Obtainx2 − y2 =8and 2xy=−15 | A1
Eliminate one unknown and find a horizontal equation in the other | M1
Obtain 4x4 −32x2 −225=0or4y4 +32y2 −225=0, or three term equivalent | A1
1
Obtain answers ± (5−3i)or equivalent
2 | A1
5
--- 7(b) ---
7(b) | Show a circle with centre2+i in a relatively correct position | B1
Show a circle with radius 2 and centre not at the origin | B1
Show line through i at an angle of1πto the real axis
4 | B1
Shade the correct region | B1
4
Question | Answer | Marks\begin{enumerate}[label=(\alph*)]
\item The complex number $u$ is given by $u = 8 - 15\text{i}$. Showing all necessary working, find the two square roots of $u$. Give answers in the form $a + ib$, where the numbers $a$ and $b$ are real and exact. [5]
\item On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities $|z - 2 - \text{i}| \leqslant 2$ and $0 \leqslant \arg(z - \text{i}) \leqslant \frac{1}{4}\pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q7 [9]}}