| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Argand diagram sketching and regions |
| Difficulty | Standard +0.3 This is a multi-part complex numbers question covering standard A-level techniques: converting to exponential form (routine), verifying a root by substitution (straightforward), and sketching a region defined by simple inequalities (circle and horizontal line). All parts are textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02d Exponential form: re^(i*theta)4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(r=2\) | B1 | Accept \(\sqrt{4}\) |
| State or imply \(\theta = \frac{1}{6}\pi\) | B1 | |
| Use a correct method for finding the modulus or the argument of \(u^4\) | M1 | Allow correct answers from correct \(u\) with minimal working shown |
| Obtain modulus \(16\) | A1 | |
| Obtain argument \(\frac{2}{3}\pi\) | A1 | Accept \(16e^{i\frac{2\pi}{3}}\) |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(u\) and carry out a correct method for finding \(u^3\) | M1 | (\(u^3=8i\)) Follow *their* \(u^3\) if found in part (i) |
| Verify \(u\) is a root of the given equation | A1 | |
| State that the other root is \(\sqrt{3}-i\) | B1 | |
| Alternative method: | ||
| State that the other root is \(\sqrt{3}-i\) | B1 | |
| Form quadratic factor and divide cubic by quadratic | M1 | \((z-\sqrt{3}-i)(z-\sqrt{3}+i)(=z^2-2\sqrt{3}z+4)\) |
| Verify that remainder is zero and hence that \(u\) is a root of the given equation | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show the point representing \(u\) in a relatively correct position | B1 | |
| Show a circle with centre \(u\) and radius 2 | B1 | FT on the point representing \(u\). Condone near miss of origin |
| Show the line \(y = 2\) | B1 | |
| Shade the correct region | B1 | |
| Show that the line and circle intersect on \(x = 0\) | B1 | Condone near miss |
| Total | 5 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $r=2$ | B1 | Accept $\sqrt{4}$ |
| State or imply $\theta = \frac{1}{6}\pi$ | B1 | |
| Use a correct method for finding the modulus or the argument of $u^4$ | M1 | Allow correct answers from correct $u$ with minimal working shown |
| Obtain modulus $16$ | A1 | |
| Obtain argument $\frac{2}{3}\pi$ | A1 | Accept $16e^{i\frac{2\pi}{3}}$ |
| **Total** | **5** | |
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $u$ and carry out a correct method for finding $u^3$ | M1 | ($u^3=8i$) Follow *their* $u^3$ if found in part (i) |
| Verify $u$ is a root of the given equation | A1 | |
| State that the other root is $\sqrt{3}-i$ | B1 | |
| **Alternative method:** | | |
| State that the other root is $\sqrt{3}-i$ | B1 | |
| Form quadratic factor and divide cubic by quadratic | M1 | $(z-\sqrt{3}-i)(z-\sqrt{3}+i)(=z^2-2\sqrt{3}z+4)$ |
| Verify that remainder is zero and hence that $u$ is a root of the given equation | A1 | |
| **Total** | **3** | |
## Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show the point representing $u$ in a relatively correct position | **B1** | |
| Show a circle with centre $u$ and radius 2 | **B1** | FT on the point representing $u$. Condone near miss of origin |
| Show the line $y = 2$ | **B1** | |
| Shade the correct region | **B1** | |
| Show that the line and circle intersect on $x = 0$ | **B1** | Condone near miss |
| **Total** | **5** | |10 Throughout this question the use of a calculator is not permitted.\\
The complex number $( \sqrt { } 3 ) + \mathrm { i }$ is denoted by $u$.\\
(i) Express $u$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$, giving the exact values of $r$ and $\theta$. Hence or otherwise state the exact values of the modulus and argument of $u ^ { 4 }$.\\
(ii) Verify that $u$ is a root of the equation $z ^ { 3 } - 8 z + 8 \sqrt { } 3 = 0$ and state the other complex root of this equation.\\
(iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - u | \leqslant 2$ and $\operatorname { Im } z \geqslant 2$, where $\operatorname { Im } z$ denotes the imaginary part of $z$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P3 2019 Q10 [13]}}