| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Square roots of complex numbers |
| Difficulty | Standard +0.3 Part (a) is a standard procedure for finding square roots of complex numbers by equating real and imaginary parts, requiring algebraic manipulation but following a well-established method. Part (b) involves sketching a standard circular region and finding an argument using basic geometry/trigonometry. Both parts are routine A-level Further Maths 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-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain \(x^2 - y^2 = 1\) and \(2xy = -2\sqrt{6}\) | M1* A1 | |
| Eliminate one variable and find an equation in the other | M1(dep*) | |
| Obtain \(x^4 - x^2 - 6 = 0\) or \(y^4 + y^2 - 6 = 0\), or 3-term equivalent | A1 | |
| Obtain answers \(\pm(\sqrt{3} - i\sqrt{2})\) | A1 | [5] |
| Answer | Marks |
|---|---|
| and find values of \(R\) and either \(\cos\theta\) or \(\sin\theta\) or \(\tan\theta\) | M1* |
| Obtain \(\pm\sqrt{5}(\cos \frac{1}{5}\theta + i\sin \frac{1}{5}\theta)\), and \(\cos\theta = \frac{1}{5}\) or \(\sin\theta = -\frac{2\sqrt{6}}{5}\) or \(\tan\theta = -2\sqrt{6}\) | A1 |
| Use correct method to find an exact value of \(\cos \frac{1}{2}\theta\) or \(\sin \frac{1}{2}\theta\) | M1(dep*) |
| Obtain \(\cos \frac{1}{2}\theta = \pm\sqrt{\frac{3}{5}}\) and \(\sin \frac{1}{2}\theta = \pm\sqrt{\frac{2}{5}}\), or equivalent | A1 |
| Obtain answers \(\pm(\sqrt{3} - i\sqrt{2})\), or equivalent | A1 |
| [Condone omission of \(\pm\) except in the final answers.] | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Show a circle with centre at the point representing \(3i\) and radius 2 | B1√ B1√ | |
| Shade the interior of the circle | B1√ | |
| Carry out a complete method for finding the greatest value of \(\arg z\) | M1 | |
| Obtain answer \(131.8°\) or \(2.30\) (or \(2.3\)) radians | A1 | [5] |
**(a) EITHER: Square $x + iy$ and equate real and imaginary parts to 1 and $-2\sqrt{6}$ respectively**
Obtain $x^2 - y^2 = 1$ and $2xy = -2\sqrt{6}$ | M1* A1 |
Eliminate one variable and find an equation in the other | M1(dep*) |
Obtain $x^4 - x^2 - 6 = 0$ or $y^4 + y^2 - 6 = 0$, or 3-term equivalent | A1 |
Obtain answers $\pm(\sqrt{3} - i\sqrt{2})$ | A1 | [5]
**OR: Denoting $1 - 2\sqrt{6}i$ by $Rcis\theta$, state, or imply, square roots are $\pm\sqrt{Rcis(\frac{1}{2}\theta)}$**
and find values of $R$ and either $\cos\theta$ or $\sin\theta$ or $\tan\theta$ | M1* |
Obtain $\pm\sqrt{5}(\cos \frac{1}{5}\theta + i\sin \frac{1}{5}\theta)$, and $\cos\theta = \frac{1}{5}$ or $\sin\theta = -\frac{2\sqrt{6}}{5}$ or $\tan\theta = -2\sqrt{6}$ | A1 |
Use correct method to find an exact value of $\cos \frac{1}{2}\theta$ or $\sin \frac{1}{2}\theta$ | M1(dep*) |
Obtain $\cos \frac{1}{2}\theta = \pm\sqrt{\frac{3}{5}}$ and $\sin \frac{1}{2}\theta = \pm\sqrt{\frac{2}{5}}$, or equivalent | A1 |
Obtain answers $\pm(\sqrt{3} - i\sqrt{2})$, or equivalent | A1 |
[Condone omission of $\pm$ except in the final answers.] | [5]
**(b) Show point representing $3i$ on a sketch of an Argand diagram**
Show a circle with centre at the point representing $3i$ and radius 2 | B1√ B1√ |
Shade the interior of the circle | B1√ |
Carry out a complete method for finding the greatest value of $\arg z$ | M1 |
Obtain answer $131.8°$ or $2.30$ (or $2.3$) radians | A1 | [5]
[The f.t. is on solutions where the centre is at the point representing $-3i$.]10
\begin{enumerate}[label=(\alph*)]
\item Showing your working, find the two square roots of the complex number $1 - ( 2 \sqrt { } 6 ) \mathrm { i }$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact.
\item On a sketch of an Argand diagram, shade the region whose points represent the complex numbers $z$ which satisfy the inequality $| z - 3 i | \leqslant 2$. Find the greatest value of $\arg z$ for points in this region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2011 Q10 [10]}}