| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Moderate -0.8 This is a routine complex numbers question testing standard techniques: conjugate notation, real part extraction, division by multiplying by conjugate, and conversion to modulus-argument form. All parts follow textbook procedures with no problem-solving or novel insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | (i) |
| [1] | 1.2 | |
| 1 | (a) | (ii) |
| so Re(iz) = –b | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (b) | (i) |
| Answer | Marks |
|---|---|
| = 1+√3i | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1a | |
| 1.1 | an intermediate step must be seen before final answer | |
| 1 | (b) | (ii) |
| Answer | Marks |
|---|---|
| (cid:2871) (cid:2871) | B1ft |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | for either their modulus or argument |
Question 1:
1 | (a) | (i) | a – ib | B1
[1] | 1.2
1 | (a) | (ii) | iz = –b +ai
so Re(iz) = –b | M1
A1
[2] | 1.1
1.1
1 | (b) | (i) | DR
5+√3i (5+√3i)(2+√3i)
=
2−√3i (2−√3i)(2+√3i)
7+7√3i
=
7
= 1+√3i | M1
A1
[2] | 1.1a
1.1 | an intermediate step must be seen before final answer
1 | (b) | (ii) | (cid:3627)1+√3i(cid:3627)= 2
𝜋
arg(1+√3i)=
3
(cid:3095) (cid:3095)
so 𝑤 = 2(cos +i sin )
(cid:2871) (cid:2871) | B1ft
B1
[2] | 1.1
1.1 | for either their modulus or argument
soi
cao. Allow 60°1
\begin{enumerate}[label=(\alph*)]
\item The complex number $\mathrm { a } + \mathrm { ib }$ is denoted by $z$.
\begin{enumerate}[label=(\roman*)]
\item Write down $z ^ { * }$.
\item Find $\operatorname { Re } ( \mathrm { iz } )$.
\end{enumerate}\item The complex number $w$ is given by $w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }$.
\begin{enumerate}[label=(\roman*)]
\item In this question you must show detailed reasoning.
Express $w$ in the form $\mathrm { x } + \mathrm { iy }$.
\item Convert $w$ to modulus-argument form.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2023 Q1 [7]}}