| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring basic complex number operations (finding 1/z using modulus/argument rules) and vector addition on an Argand diagram to find the fourth vertex of a parallelogram. While it involves multiple steps, each is routine: reciprocal gives modulus 1/2 and argument -π/3, then w = z + 1/z is simple addition. No novel insight required, just careful application of standard techniques. |
| 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 |
|---|---|---|
| 7 | (a) | A |
| Answer | Marks |
|---|---|
| B | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | A approx. 60 to real axis or 1 + i3 indicated |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (b) | |
| Answer | Marks |
|---|---|
| z 2 3 3 4 | M1 |
| Answer | Marks |
|---|---|
| B1 | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | converting to a + bi form |
| Answer | Marks |
|---|---|
| 5 | M1 |
| Answer | Marks |
|---|---|
| B1 | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | or equivalent methods (e.g. vector displacements) |
| Answer | Marks | Guidance |
|---|---|---|
| Equation of BC is y = 3 x −3/2 | M1 | equations of BC and AC both correctly calculated |
| Answer | Marks |
|---|---|
| Solving simultaneously x = 5/4, y = 33/4 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 6 1 6 2 | B1 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | B1 | 0.80 or better or 46 or better, must be from correct w |
Question 7:
7 | (a) | A
2
O
½
B | B1
B1
[2] | 1.1
1.1 | A approx. 60 to real axis or 1 + i3 indicated
B approx. 60 to real axis and OB = ¼ OA or ¼ (1 − i3) indicated
7 | (b) |
z 2 ( c o s i s in ) = +
3 3
= 1 + 3 i
1 1 1
( c o s i s in ) (1 3 i) = − + − = −
z 2 3 3 4 | M1
A1
B1 | 3.1a
1.1
1.1 | converting to a + bi form
NB these first three marks may be awarded if gained in part (a)
1
w = z +
z
5 3 3
= + i
4 4
2 5 2 7 1 3
w = + =
1 6 1 6 2
3 3
a r g ( w ) = a r c ta n = 0 .8 0 5
5 | M1
A1
B1
B1 | 3.1a
1.1ft
3.2a
3.2a | or equivalent methods (e.g. vector displacements)
5 3 3
o r ,
4 4
must be from correct w
0.80 or better or 46 or better, must be from correct w
Alternative solution
Equation of BC is y = 3 x −3/2 | M1 | equations of BC and AC both correctly calculated
Equation of AC is y = −3 x + 23
Solving simultaneously x = 5/4, y = 33/4 | A1
2 5 2 7 1 3
w = + =
1 6 1 6 2 | B1 | B1 | must be from correct w | must be from correct w
3 3
a r g ( w ) = a r c ta n = 0 .8 0 5
5 | B1 | 0.80 or better or 46 or better, must be from correct w
[7]7 On an Argand diagram, the point A represents the complex number $z$ with modulus 2 and argument $\frac { 1 } { 3 } \pi$. The point B represents $\frac { 1 } { z }$.
\begin{enumerate}[label=(\alph*)]
\item Sketch an Argand diagram showing the origin O and the points A and B .
\item The point C is such that OACB is a parallelogram. C represents the complex number $w$.
Determine each of the following.
\begin{itemize}
\item The modulus of $w$, giving your answer in exact form.
\item The argument of $w$, giving your answer correct to $\mathbf { 3 }$ significant figures.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2022 Q7 [9]}}