| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Complex number loci on Argand diagrams |
| Difficulty | Standard +0.3 This is a straightforward multi-part Further Maths question testing standard complex number operations (addition, multiplication, division by multiplying by conjugate), modulus/argument calculations, and sketching basic loci (circle and half-line). While Further Maths content is inherently harder, these are routine textbook exercises requiring only direct application of formulas with no problem-solving insight, placing it slightly above average difficulty overall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\alpha + \beta = 1 + j\) | B1 | |
| \(\alpha\beta = (2-j)(-1+2j) = -2+4j+j-2j^2 = 5j\) | M1, A1 | |
| \(\frac{\alpha}{\beta} = \frac{(2-j)(-1-2j)}{(-1+2j)(-1-2j)} = \frac{-2-4j+j+2j^2}{5} = -\frac{4}{5} - \frac{3}{5}j\) or \(\frac{-4-3j}{5}\) | M1, A1, A1, [6] | Use of conjugate of denominator |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(r = | \alpha | = \sqrt{5}\) |
| \(\theta = \arg\alpha = -0.464\) | B1, [2] | Accept degree equivalent (\(-26.6°\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Circle, centre \(2-j\), radius \(2\) | B1, B1, [2] | Argand diagram with circle. 1 mark for centre, one mark for radius |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Half line from \(-1+2j\), making an angle of \(\frac{\pi}{4}\) to the positive real axis | B1, B1, [2] | Argand diagram with half line. One mark for angle |
# Question 8:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\alpha + \beta = 1 + j$ | B1 | |
| $\alpha\beta = (2-j)(-1+2j) = -2+4j+j-2j^2 = 5j$ | M1, A1 | |
| $\frac{\alpha}{\beta} = \frac{(2-j)(-1-2j)}{(-1+2j)(-1-2j)} = \frac{-2-4j+j+2j^2}{5} = -\frac{4}{5} - \frac{3}{5}j$ or $\frac{-4-3j}{5}$ | M1, A1, A1, **[6]** | Use of conjugate of denominator |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $r = |\alpha| = \sqrt{5}$ | B1 | |
| $\theta = \arg\alpha = -0.464$ | B1, **[2]** | Accept degree equivalent ($-26.6°$) |
## Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Circle, centre $2-j$, radius $2$ | B1, B1, **[2]** | Argand diagram with circle. 1 mark for centre, one mark for radius |
## Part (iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| Half line from $-1+2j$, making an angle of $\frac{\pi}{4}$ to the positive real axis | B1, B1, **[2]** | Argand diagram with half line. One mark for angle |8 Two complex numbers are given by $\alpha = 2 - \mathrm { j }$ and $\beta = - 1 + 2 \mathrm { j }$.\\
(i) Find $\alpha + \beta , \alpha \beta$ and $\frac { \alpha } { \beta }$ in the form $a + b \mathrm { j }$, showing your working.\\
(ii) Find the modulus of $\alpha$, leaving your answer in surd form. Find also the argument of $\alpha$.\\
(iii) Sketch the locus $| z - \alpha | = 2$ on an Argand diagram.\\
(iv) On a separate Argand diagram, sketch the locus $\arg ( z - \beta ) = \frac { 1 } { 4 } \pi$.
\hfill \mbox{\textit{OCR MEI FP1 2005 Q8 [12]}}