| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus and argument with operations |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing standard complex number operations (modulus, argument, multiplication, division) and Argand diagram plotting. While it's multi-part and involves division requiring rationalization of the denominator, all techniques are routine applications of FP1 content with no novel problem-solving required. The values are chosen to give clean answers (α has modulus 2, argument π/6). Slightly easier than average A-level due to its procedural nature, but the Further Maths context and multiple parts keep it close to baseline. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\arg\alpha = \frac{\pi}{6},\ | \alpha | = 2\) |
| \(\arg\beta = \frac{\pi}{2},\ | \beta | = 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\alpha\beta = (\sqrt{3}+j)3j = -3 + 3\sqrt{3}j\) | M1, A1 | Use of \(j^2 = -1\); correct |
| \(\frac{\beta}{\alpha} = \frac{3j}{\sqrt{3}+j} = \frac{3j(\sqrt{3}-j)}{(\sqrt{3}+j)(\sqrt{3}-j)}\) | M1 | Correct use of conjugate of denominator |
| \(= \frac{3+3\sqrt{3}j}{4} = \frac{3}{4} + \frac{3\sqrt{3}}{4}j\) | A1, A1 [5] | Denominator \(= 4\); all correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Argand diagram showing \(\alpha\), \(\beta\), \(\alpha\beta\), \(\frac{\beta}{\alpha}\) plotted | M1, A1(ft) [2] | Argand diagram with at least one correct point; correct relative positions with appropriate labelling |
# Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\arg\alpha = \frac{\pi}{6},\ |\alpha| = 2$ | B1, B1 | Modulus of $\alpha$; argument of $\alpha$ (allow $30°$) |
| $\arg\beta = \frac{\pi}{2},\ |\beta| = 3$ | B1 **[3]** | Both modulus and argument of $\beta$ (allow $90°$) |
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# Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\alpha\beta = (\sqrt{3}+j)3j = -3 + 3\sqrt{3}j$ | M1, A1 | Use of $j^2 = -1$; correct |
| $\frac{\beta}{\alpha} = \frac{3j}{\sqrt{3}+j} = \frac{3j(\sqrt{3}-j)}{(\sqrt{3}+j)(\sqrt{3}-j)}$ | M1 | Correct use of conjugate of denominator |
| $= \frac{3+3\sqrt{3}j}{4} = \frac{3}{4} + \frac{3\sqrt{3}}{4}j$ | A1, A1 **[5]** | Denominator $= 4$; all correct |
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# Question 8(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Argand diagram showing $\alpha$, $\beta$, $\alpha\beta$, $\frac{\beta}{\alpha}$ plotted | M1, A1(ft) **[2]** | Argand diagram with at least one correct point; correct relative positions with appropriate labelling |8 Two complex numbers, $\alpha$ and $\beta$, are given by $\alpha = \sqrt { 3 } + \mathrm { j }$ and $\beta = 3 \mathrm { j }$.\\
(i) Find the modulus and argument of $\alpha$ and $\beta$.\\
(ii) Find $\alpha \beta$ and $\frac { \beta } { \alpha }$, giving your answers in the form $a + b \mathrm { j }$, showing your working.\\
(iii) Plot $\alpha , \beta , \alpha \beta$ and $\frac { \beta } { \alpha }$ on a single Argand diagram.
\hfill \mbox{\textit{OCR MEI FP1 2010 Q8 [10]}}