| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.3 Part (a) is a standard complex number manipulation requiring division (multiply by conjugate), conversion to modulus-argument form, and calculator work—routine for A-level Further Maths. Part (b) involves recognizing a circle locus and finding the minimum distance from origin, which requires geometric insight but is a common exam pattern. The question tests fundamental techniques without requiring novel problem-solving approaches. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| 8(i) | EITHER: Multiply numerator and denominator by 1 + 2i, or equivalent, or equate to x |
| + iy, obtain two equations in x and y and solve for x or for y | M1 |
| Answer | Marks |
|---|---|
| 5 5 | A1 |
| Use correct method to find either r or θ | M1 |
| Obtain r = 1.61 | A1 |
| Obtain θ = 2.09 | A1 |
| OR: Find modulus or argument of 2 + 3i or of 1 – 2i | B1 |
| Use correct method to find r | M1 |
| Obtain r = 1.61 | A1 |
| Use correct method to find θ | M1 |
| Obtain θ = 2.09 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(ii) | Show a circle with centre 3 – 2i | B1 |
| Show a circle with radius 1 | B1ft | Centre not at the origin |
| Carry out a correct method for finding the least value of z | M1 | |
| Obtain answer 13 – 1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 8:
--- 8(i) ---
8(i) | EITHER: Multiply numerator and denominator by 1 + 2i, or equivalent, or equate to x
+ iy, obtain two equations in x and y and solve for x or for y | M1
4 7
Obtain quotient − + i, or equivalent
5 5 | A1
Use correct method to find either r or θ | M1
Obtain r = 1.61 | A1
Obtain θ = 2.09 | A1
OR: Find modulus or argument of 2 + 3i or of 1 – 2i | B1
Use correct method to find r | M1
Obtain r = 1.61 | A1
Use correct method to find θ | M1
Obtain θ = 2.09 | A1
5
--- 8(ii) ---
8(ii) | Show a circle with centre 3 – 2i | B1
Show a circle with radius 1 | B1ft | Centre not at the origin
Carry out a correct method for finding the least value of z | M1
Obtain answer 13 – 1 | A1
4
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\alph*)]
\item Showing all necessary working, express the complex number $\frac{2 + 3i}{1 - 2i}$ in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leqslant \pi$. Give the values of $r$ and $\theta$ correct to 3 significant figures. [5]
\item On an Argand diagram sketch the locus of points representing complex numbers $z$ satisfying the equation $|z - 3 + 2i| = 1$. Find the least value of $|z|$ for points on this locus, giving your answer in an exact form. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q8 [9]}}