| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Argument calculations and identities |
| Difficulty | Challenging +1.2 This is a multi-step Further Maths question requiring conversion between forms, complex multiplication, and using De Moivre's theorem to find an exact trigonometric value. While it involves several techniques, each step follows standard procedures with clear signposting. The final trigonometric extraction requires some insight but is a well-known application type in Further Maths complex numbers. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02f Convert between forms: cartesian and modulus-argument4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (i) |
| Answer | Marks |
|---|---|
| 1 4 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | 1.41 or better |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (ii) |
| Answer | Marks |
|---|---|
| = 3+i | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.1 | 1.73 or better | |
| 7 | (b) | z z =(1+i)( 3+i) |
| Answer | Marks |
|---|---|
| 4 4 | M1 |
| Answer | Marks |
|---|---|
| [5] | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | Finding z z (using cartesian form) |
Question 7:
7 | (a) | (i) | | z |= 2
1
arg(z )= 1π
1 4 | B1
B1
[2] | 1.1
1.1 | 1.41 or better
allow 45°
7 | (a) | (ii) | z =2 ( cos1π+isin1π )
2 6 6
= 3+i | M1
A1
[2] | 1.1
1.1 | 1.73 or better
7 | (b) | z z =(1+i)( 3+i)
1 2
= 3−1+( 3+1)i
z z =2 2 ( cos 5π+isin 5π )
1 2 12 12
3+1
So sin 5π=
12 2 2
2( 3+1) 6+ 2
= =
4 4 | M1
A1
M1
A1
A1
[5] | 3.1a
1.1
3.1a
1.1
3.2a | Finding z z (using cartesian form)
1 2
Use of arg(z z ) = arg z + argz
1 2 1 27
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the modulus and argument of $z _ { 1 }$, where $z _ { 1 } = 1 + \mathrm { i }$.
\item Given that $\left| z _ { 2 } \right| = 2$ and $\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi$, express $z _ { 2 }$ in a + bi form, where $a$ and $b$ are exact real numbers.
\end{enumerate}\item Using these results, find the exact value of $\sin \frac { 5 } { 12 } \pi$, giving the answer in the form $\frac { \sqrt { m } + \sqrt { n } } { p }$, where $m , n$ and $p$ are integers.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q7 [9]}}