| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of unity applications |
| Difficulty | Standard +0.3 This is a structured multi-part question on standard FP2 topics (exponential form, roots of unity, rotations). Part (i) is trivial recall, parts (ii-iii) are routine applications of exponential form and nth roots, parts (iv-v) involve straightforward manipulation. While it requires multiple techniques, each step follows standard procedures without requiring novel insight or complex problem-solving. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Modulus \(= 1\) | B1 | Must be separate |
| Argument \(= \dfrac{\pi}{3}\) | B1 | Accept \(60°\), \(1.05^c\) |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Diagram with points in correct quadrants | G2,1,0 | A in first quadrant, argument \(\approx\dfrac{\pi}{4}\); B in second quadrant, same mod; B′ in fourth quadrant, same mod; Symmetry. G1: 3 points and at least 2 of above, or B, B′ on axes, or BOB′ straight line, or BOB′ reflex |
| \(a = 2e^{\frac{j\pi}{4}}\) | B1 | Must be in required form (accept \(r=2\), \(\theta=\pi/4\)) |
| \(\arg b = \dfrac{\pi}{4} \pm \dfrac{\pi}{3}\) | M1 | Rotate by adding (or subtracting) \(\pi/3\) to (or from) argument. Must be \(\pi/3\) |
| \(b = 2e^{\frac{j\pi}{12}},\ 2e^{\frac{7j\pi}{12}}\) | A1ft | Both. Ft value of \(r\) for \(a\). Must be in required form, but don't penalise twice |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(z_1^6 = \left(\sqrt{2}e^{\frac{j\pi}{3}}\right)^6 = \left(\sqrt{2}\right)^6 e^{2j\pi}\) | M1 | \(\left(\sqrt{2}\right)^6 = 8\) or \(\dfrac{\pi}{3}\times 6 = 2\pi\) seen |
| \(= 8\) | A1 (ag) | www |
| Others are \(re^{j\theta}\) where \(r = \sqrt{2}\) | M1 | "Add" \(\dfrac{\pi}{3}\) to argument more than once |
| \(\theta = -\dfrac{2\pi}{3},\ -\dfrac{\pi}{3},\ 0,\ \dfrac{2\pi}{3},\ \pi\) | A1 | Correct constant \(r\) and five values of \(\theta\). Accept \(\theta\) in \([0, 2\pi]\) or in degrees |
| Diagram: regular hexagon | G1 | 6 points on vertices of regular hexagon |
| Correctly positioned | G1 | 2 roots on real axis. Ignore scales. SC1 if G0 and 5 points correctly plotted |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(w = z_1 e^{\frac{j\pi}{12}} = \sqrt{2}e^{\frac{j\pi}{3}}e^{-\frac{j\pi}{12}} = \sqrt{2}e^{\frac{j\pi}{4}}\) | M1 | \(\arg w = \dfrac{\pi}{3} - \dfrac{\pi}{12}\) |
| \(= \sqrt{2}\left(\cos\dfrac{\pi}{4} + j\sin\dfrac{\pi}{4}\right)\) | ||
| \(= 1 + j\) | A1 | Or B2 |
| Same modulus as \(z_1\) | G1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(w^6 = \left(\sqrt{2}e^{\frac{j\pi}{4}}\right)^6 = 8e^{\frac{3j\pi}{2}}\) | M1 | Or \(z_1^6 e^{-\frac{j\pi}{2}} = 8e^{-\frac{j\pi}{2}}\) |
| \(= -8j\) | A1 | cao. Evaluated |
| Total | 2 |
# Question 2:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Modulus $= 1$ | B1 | Must be separate |
| Argument $= \dfrac{\pi}{3}$ | B1 | Accept $60°$, $1.05^c$ |
| **Total** | **2** | |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Diagram with points in correct quadrants | G2,1,0 | A in first quadrant, argument $\approx\dfrac{\pi}{4}$; B in second quadrant, same mod; B′ in fourth quadrant, same mod; Symmetry. G1: 3 points and at least 2 of above, or B, B′ on axes, or BOB′ straight line, or BOB′ reflex |
| $a = 2e^{\frac{j\pi}{4}}$ | B1 | Must be in required form (accept $r=2$, $\theta=\pi/4$) |
| $\arg b = \dfrac{\pi}{4} \pm \dfrac{\pi}{3}$ | M1 | Rotate by adding (or subtracting) $\pi/3$ to (or from) argument. Must be $\pi/3$ |
| $b = 2e^{\frac{j\pi}{12}},\ 2e^{\frac{7j\pi}{12}}$ | A1ft | Both. Ft value of $r$ for $a$. Must be in required form, but don't penalise twice |
| **Total** | **5** | |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $z_1^6 = \left(\sqrt{2}e^{\frac{j\pi}{3}}\right)^6 = \left(\sqrt{2}\right)^6 e^{2j\pi}$ | M1 | $\left(\sqrt{2}\right)^6 = 8$ or $\dfrac{\pi}{3}\times 6 = 2\pi$ seen |
| $= 8$ | A1 (ag) | www |
| Others are $re^{j\theta}$ where $r = \sqrt{2}$ | M1 | "Add" $\dfrac{\pi}{3}$ to argument more than once |
| $\theta = -\dfrac{2\pi}{3},\ -\dfrac{\pi}{3},\ 0,\ \dfrac{2\pi}{3},\ \pi$ | A1 | Correct constant $r$ and five values of $\theta$. Accept $\theta$ in $[0, 2\pi]$ or in degrees |
| Diagram: regular hexagon | G1 | 6 points on vertices of regular hexagon |
| Correctly positioned | G1 | 2 roots on real axis. Ignore scales. SC1 if G0 and 5 points correctly plotted |
| **Total** | **6** | |
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| $w = z_1 e^{\frac{j\pi}{12}} = \sqrt{2}e^{\frac{j\pi}{3}}e^{-\frac{j\pi}{12}} = \sqrt{2}e^{\frac{j\pi}{4}}$ | M1 | $\arg w = \dfrac{\pi}{3} - \dfrac{\pi}{12}$ |
| $= \sqrt{2}\left(\cos\dfrac{\pi}{4} + j\sin\dfrac{\pi}{4}\right)$ | | |
| $= 1 + j$ | A1 | Or B2 |
| Same modulus as $z_1$ | G1 | |
| **Total** | **3** | |
## Part (v)
| Answer | Mark | Guidance |
|--------|------|----------|
| $w^6 = \left(\sqrt{2}e^{\frac{j\pi}{4}}\right)^6 = 8e^{\frac{3j\pi}{2}}$ | M1 | Or $z_1^6 e^{-\frac{j\pi}{2}} = 8e^{-\frac{j\pi}{2}}$ |
| $= -8j$ | A1 | cao. Evaluated |
| **Total** | **2** | |
---(i) Write down the modulus and argument of the complex number $\mathrm { e } ^ { \mathrm { j } \pi / 3 }$.\\
(ii) The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number $a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }$ corresponds to the complex number $b$.
Show A and the two possible positions for B in a sketch. Express $a$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$. Find the two possibilities for $b$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$.\\
(iii) Given that $z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }$, show that $z _ { 1 } ^ { 6 } = 8$. Write down, in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, the other five complex numbers $z$ such that $z ^ { 6 } = 8$. Sketch all six complex numbers in a new Argand diagram.
Let $w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }$.\\
(iv) Find $w$ in the form $x + \mathrm { j } y$, and mark this complex number on your Argand diagram.\\
(v) Find $w ^ { 6 }$, expressing your answer in as simple a form as possible.
\hfill \mbox{\textit{OCR MEI FP2 2009 Q2 [18]}}