| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.8 This is a Further Maths question requiring geometric interpretation of complex number multiplication by e^(iθ) as rotation, understanding of isosceles triangles in the Argand diagram, and applying rotation transformations to find specific complex numbers. Part (i) is straightforward recognition that multiplication by e^(iπ/6) rotates by π/6 with preserved modulus. Part (ii) requires translating the geometric constraint into algebraic form: rotating (5+2i)-(1+i) by ±π/6 about point C, which involves multiple steps and careful calculation. This is moderately challenging for Further Maths but follows standard techniques. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02m Geometrical effects: multiplication and division4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Diagram | B1 | must have triangle where B is anticlockwise from A, looks isosceles, \(AOB < \frac{\pi}{4}\), if axes labelled then must be correct |
| \(OB = | z e^{i\pi/1} | = |
| So triangle is isosceles oe \(\angle AOB = \frac{\pi}{6}\) | A1 | without contradictions |
| B1 | or 30° | |
| [4] | ||
| (ii) \(w = (1+i) + ((5+2i) - (1+i))e^{\pm i\pi/1}\) | M1 | Rotation of CD |
| \(w = \frac{1}{2} + 2\sqrt{3} + (3 + \frac{1}{2}\sqrt{3})i\) or \(\frac{1}{2} + 2\sqrt{3} + (-1 + \frac{1}{2}\sqrt{3})i\) | M1 | converts \(e^{\pm i\pi/1}\) into \(a + bi\) form |
| A1 | ||
| [5] | Condone omission of \(\pm\) in M marks | |
| Alternative method: \(CE = \begin{pmatrix}a\\b\end{pmatrix}\), \(CD = \begin{pmatrix}4\\1\end{pmatrix}\). Now use \(CE \cdot CD = 17\cos(\pi/6)\) and \(CE^2 = 17\) to obtain equations \(4a + b = 17\sqrt{3}/2\) and \(a^2 + b^2 = 17\) (or equivalent) | M1 | (for both). |
| Obtain 3-term quadratic in one variable and solve to get one correct value of \(a\) or \(b\) | M1 | Quadratics are \(a^2 - 4\sqrt{3}a + 47/4 = 0\) and \(b^2 - \sqrt{3}b - 13/4 = 0\) |
| \((a,b) = (2\sqrt{3} \pm \frac{1}{2}, \frac{1}{2}\sqrt{3} \pm 2)\) Final answer | A1 A1 |
**(i)** Diagram | B1 | must have triangle where B is anticlockwise from A, looks isosceles, $AOB < \frac{\pi}{4}$, if axes labelled then must be correct
$OB = |z e^{i\pi/1}| = |z||e^{i\pi/1}| = |z|.1 = |z| = OA$ | M1 | condone $OB = |z| = OA$
So triangle is isosceles oe $\angle AOB = \frac{\pi}{6}$ | A1 | without contradictions
| B1 | or 30°
| [4] |
**(ii)** $w = (1+i) + ((5+2i) - (1+i))e^{\pm i\pi/1}$ | M1 | Rotation of CD
$w = \frac{1}{2} + 2\sqrt{3} + (3 + \frac{1}{2}\sqrt{3})i$ or $\frac{1}{2} + 2\sqrt{3} + (-1 + \frac{1}{2}\sqrt{3})i$ | M1 | converts $e^{\pm i\pi/1}$ into $a + bi$ form
| A1 |
| [5] | Condone omission of $\pm$ in M marks
Alternative method: $CE = \begin{pmatrix}a\\b\end{pmatrix}$, $CD = \begin{pmatrix}4\\1\end{pmatrix}$. Now use $CE \cdot CD = 17\cos(\pi/6)$ and $CE^2 = 17$ to obtain equations $4a + b = 17\sqrt{3}/2$ and $a^2 + b^2 = 17$ (or equivalent) | M1 | (for both).
Obtain 3-term quadratic in one variable and solve to get one correct value of $a$ or $b$ | M1 | Quadratics are $a^2 - 4\sqrt{3}a + 47/4 = 0$ and $b^2 - \sqrt{3}b - 13/4 = 0$
$(a,b) = (2\sqrt{3} \pm \frac{1}{2}, \frac{1}{2}\sqrt{3} \pm 2)$ Final answer | A1 A1 |4 In an Argand diagram, the complex numbers $0 , z$ and $z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }$ are represented by the points $O , A$ and $B$ respectively.\\
(i) Sketch a possible Argand diagram showing the triangle $O A B$. Show that the triangle is isosceles and state the size of angle $A O B$.
The complex numbers $1 + \mathrm { i }$ and $5 + 2 \mathrm { i }$ are represented by the points $C$ and $D$ respectively. The complex number $w$ is represented by the point $E$, such that $C D = C E$ and angle $D C E = \frac { 1 } { 6 } \pi$.\\
(ii) Calculate the possible values of $w$, giving your answers exactly in the form $a + b \mathrm { i }$.
\hfill \mbox{\textit{OCR FP3 2015 Q4 [9]}}