| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic complex number operations: multiplication, modulus-argument form conversion, and verification of the argument addition property. While it requires multiple steps and 'detailed reasoning', each part uses standard techniques with no novel insight needed. It's slightly above average difficulty (0.0) because it's Further Maths content and requires careful calculation across three parts, but remains routine for this level. |
| 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, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| DR: \(z_1z_2 = (3+4i)(-5+12i) = -15 + 36i - 20i - 48\) | M1 | Attempt at expansion (4 terms soi) using \(i^2 = -1\). DR – Need to see at least one line of expanded terms before answer. |
| \(= -63 + 16i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| DR: \( | z_2 | = \sqrt{(-5)^2 + 12^2} = \sqrt{169} = 13\) |
| \(\tan^{-1}\!\left(\dfrac{12}{-5}\right)\) | M1 | Evidence of using trigonometry towards finding the correct angle, perhaps by finding a related angle. Treat \(\tan^{-1}\!\left(\dfrac{12}{5}\right)\) as such evidence for M1 but not \(\tan^{-1}\!\left(-\dfrac{5}{12}\right)\) or \(\tan^{-1}\!\left(\dfrac{5}{12}\right)\) unless supported e.g. by a diagram or by working leading to correct answer. |
| \(\therefore z_2 = 13(\cos1.97 + i\sin1.97)\) (3 sf) | A1 | For argument accept awrt 1.97 only. Do not accept answers not written correctly in mod-arg form. Do not accept \(-1.18\) or \(-4.32\) as argument. Answer must be in radians for A1. Accept \([r, \theta]\) or \(r\operatorname{cis}\theta\). Note: is \(1.965587446\ldots\) e.g. do not accept \(13\cos1.97 + 13i\sin1.97\); \(13(\cos4.32 - i\sin4.32)\). NB \(z_1 = 5(\cos0.927 + i\sin0.927)\); \(z_1z_2 = 65(\cos2.89 + i\sin2.89)\). |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| DR: $z_1z_2 = (3+4i)(-5+12i) = -15 + 36i - 20i - 48$ | **M1** | Attempt at expansion (4 terms soi) using $i^2 = -1$. DR – Need to see at least one line of expanded terms before answer. |
| $= -63 + 16i$ | **A1** | |
**[2]**
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## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| DR: $|z_2| = \sqrt{(-5)^2 + 12^2} = \sqrt{169} = 13$ | **B1** | Not $\pm$ unless later corrected. Allow modulus of 13 for the B1 as long as no incorrect working. Treat attempt to write $z_1$ or $z_1z_2$ in mod/arg form as MR so **B0M1A1** available. |
| $\tan^{-1}\!\left(\dfrac{12}{-5}\right)$ | **M1** | Evidence of using trigonometry towards finding the correct angle, perhaps by finding a related angle. Treat $\tan^{-1}\!\left(\dfrac{12}{5}\right)$ as such evidence for M1 but not $\tan^{-1}\!\left(-\dfrac{5}{12}\right)$ or $\tan^{-1}\!\left(\dfrac{5}{12}\right)$ unless supported e.g. by a diagram or by working leading to correct answer. |
| $\therefore z_2 = 13(\cos1.97 + i\sin1.97)$ (3 sf) | **A1** | For argument accept awrt 1.97 only. Do not accept answers not written correctly in mod-arg form. Do not accept $-1.18$ or $-4.32$ as argument. Answer must be in radians for A1. Accept $[r, \theta]$ or $r\operatorname{cis}\theta$. Note: is $1.965587446\ldots$ e.g. do not accept $13\cos1.97 + 13i\sin1.97$; $13(\cos4.32 - i\sin4.32)$. NB $z_1 = 5(\cos0.927 + i\sin0.927)$; $z_1z_2 = 65(\cos2.89 + i\sin2.89)$. |
**[3]**3 In this question you must show detailed reasoning.
In this question the principal argument of a complex number lies in the interval $[ 0,2 \pi )$.\\
Complex numbers $z _ { 1 }$ and $z _ { 2 }$ are defined by $z _ { 1 } = 3 + 4 \mathrm { i }$ and $z _ { 2 } = - 5 + 12 \mathrm { i }$.
\begin{enumerate}[label=(\alph*)]
\item Determine $z _ { 1 } z _ { 2 }$, giving your answer in the form $a + b \mathrm { i }$.
\item Express $z _ { 2 }$ in modulus-argument form.
\item Verify, by direct calculation, that $\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2023 Q3 [8]}}