| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Pure square root finding |
| Difficulty | Standard +0.3 Part (a) is a standard algebraic method for finding square roots of complex numbers by equating real and imaginary parts, requiring careful manipulation but following a well-established procedure. Part (b) involves sketching a standard locus region combining a circle, argument inequality, and horizontal line—routine for P3/Further Pure but requiring accuracy. Both parts are textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Square \(a + ib\) and equate real and imaginary parts to \(-3\) and \(-2\sqrt{10}\) respectively | \*M1 | |
| Obtain \(a^2 - b^2 = -3\) and \(2ab = -2\sqrt{10}\) | A1 | |
| Eliminate one unknown and find an equation in the other | DM1 | |
| Obtain \(a^4 + 3a^2 - 10 = 0\), or \(b^4 - 3b^2 - 10 = 0\), or horizontal 3-term equivalent | A1 | |
| Obtain answers \(\pm\left(\sqrt{2} - \sqrt{5}\,\text{i}\right)\), or exact equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show point representing \(3 + \text{i}\) in relatively correct position | B1 | |
| Show a circle with radius \(3\) and centre not at the origin | B1 | |
| Show correct half line from the origin at \(\dfrac{1}{4}\pi\) to the real axis | B1 | |
| Show horizontal line \(y = 2\) | B1 | |
| Shade the correct region | B1 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Square $a + ib$ and equate real and imaginary parts to $-3$ and $-2\sqrt{10}$ respectively | \*M1 | |
| Obtain $a^2 - b^2 = -3$ and $2ab = -2\sqrt{10}$ | A1 | |
| Eliminate one unknown and find an equation in the other | DM1 | |
| Obtain $a^4 + 3a^2 - 10 = 0$, or $b^4 - 3b^2 - 10 = 0$, or horizontal 3-term equivalent | A1 | |
| Obtain answers $\pm\left(\sqrt{2} - \sqrt{5}\,\text{i}\right)$, or exact equivalent | A1 | |
**Total: 5 marks**
---
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show point representing $3 + \text{i}$ in relatively correct position | B1 | |
| Show a circle with radius $3$ and centre not at the origin | B1 | |
| Show correct half line from the origin at $\dfrac{1}{4}\pi$ to the real axis | B1 | |
| Show horizontal line $y = 2$ | B1 | |
| Shade the correct region | B1 | |
**Total: 5 marks**10
\begin{enumerate}[label=(\alph*)]
\item The complex number $u$ is given by $u = - 3 - ( 2 \sqrt { } 10 )$ i. Showing all necessary working and without using a calculator, find the square roots of $u$. Give your answers in the form $a + \mathrm { i } b$, where the numbers $a$ and $b$ are real and exact.
\item On a sketch of an Argand diagram shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi$ and $\operatorname { Im } z \geqslant 2$, where $\operatorname { Im } z$ denotes the imaginary part of the complex number $z$.\\[0pt]
[5]
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2019 Q10 [10]}}