| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus-argument form conversions |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question testing basic conversions between modulus-argument and Cartesian forms, plus standard Argand diagram plotting. Part (i) requires routine application of z = r(cos θ + j sin θ) with standard angle values, and part (ii) is simple vector addition/subtraction visualization. No problem-solving or novel insight required—purely procedural recall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| \(z_2 = 5\left(\cos\frac{\pi}{4} + j\sin\frac{\pi}{4}\right)\) | M1 | May be implied |
| \(= \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}j\) | A1 | oe (exact numerical form) |
| Answer | Marks | Guidance |
|---|---|---|
| \(z_1 + z_2 = 3 + \frac{5\sqrt{2}}{2} + \left(-2 + \frac{5\sqrt{2}}{2}\right)j = 6.54 + 1.54j\) | M1 | Attempt to add and subtract \(z_1\) and their \(z_2\) — may be implied by Argand diagram |
| Answer | Marks | Guidance |
|---|---|---|
| [Argand diagram with \(z_2\), \(z_1 + z_2\), \(z_1\), \(z_1 - z_2\) plotted] | B3 | For points cao, \(-1\) each error — dotted lines not needed |
## Question 4:
### Part (i):
$z_2 = 5\left(\cos\frac{\pi}{4} + j\sin\frac{\pi}{4}\right)$ | M1 | May be implied
$= \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}j$ | A1 | oe (exact numerical form)
**[2]**
### Part (ii):
$z_1 + z_2 = 3 + \frac{5\sqrt{2}}{2} + \left(-2 + \frac{5\sqrt{2}}{2}\right)j = 6.54 + 1.54j$ | M1 | Attempt to add and subtract $z_1$ and their $z_2$ — may be implied by Argand diagram
$z_1 - z_2 = 3 - \frac{5\sqrt{2}}{2} + \left(-2 - \frac{5\sqrt{2}}{2}\right)j = -0.54 - 5.54j$
[Argand diagram with $z_2$, $z_1 + z_2$, $z_1$, $z_1 - z_2$ plotted] | B3 | For points cao, $-1$ each error — dotted lines not needed
**[4]**
---4 The complex number $z _ { 1 }$ is $3 - 2 \mathrm { j }$ and the complex number $z _ { 2 }$ has modulus 5 and argument $\frac { \pi } { 4 }$.\\
(i) Express $z _ { 2 }$ in the form $a + b \mathrm { j }$, giving $a$ and $b$ in exact form.\\
(ii) Represent $z _ { 1 } , z _ { 2 } , z _ { 1 } + z _ { 2 }$ and $z _ { 1 } - z _ { 2 }$ on a single Argand diagram.
\hfill \mbox{\textit{OCR MEI FP1 2013 Q4 [6]}}