| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Standard +0.8 This is a Further Maths question requiring conversion to modulus-argument form, finding cube roots using De Moivre's theorem, and geometric reasoning about symmetry. While the calculation is systematic, it requires multiple steps (simplifying √48, finding modulus and argument, applying the nth root formula correctly, then geometric insight about rotational symmetry). The symmetry part requires understanding that cube roots are equally spaced by 2π/3 and identifying lines through the origin and vertices/midpoints. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (a) | DR |
| Answer | Marks |
|---|---|
| 2e9 , 2e9 and 2e 9 | M1 |
| Answer | Marks |
|---|---|
| [6] | 2.1 |
| Answer | Marks |
|---|---|
| 1.1 | Correct use of relevant |
| Answer | Marks |
|---|---|
| or eg 2e9 , 2e9 and 2e9 | Correct answer with no working: |
Question 8:
8 | (a) | DR
r2 =(−4)2+ ( 48 )2 or (rcosθ = –4 and
rsinθ = √48) or tanθ = –√3 oe
r = 8 (ie z = 8eiθ) θ = 2π/3 (ie z = rei2π/3)
3 8 or 2
2π
soi
9
2π
+2πk for k = 1 and 2 oe seen
3
2 8 4
πi πi − πi
2e9 , 2e9 and 2e 9 | M1
A1
B1ft
B1ft
M1
A1
[6] | 2.1
1.1
2.1
2.1
2.2a
1.1 | Correct use of relevant
formula(e). Some working must
be seen.
Not ±8 unless later corrected
Modulus of cube root(s) is the
cube root of their modulus
Argument of (principal) cube
root is one third of their argument
Considering further arguments at
angular distance 2π
2 8 14
πi πi πi
or eg 2e9 , 2e9 and 2e9 | Correct answer with no working:
M0A0
or eg θ = 8π/3
Must be in exponential form, not
just r = and θ =. Do not condone
any missing i’s.8 In this question you must show detailed reasoning.
The complex number $- 4 + i \sqrt { 48 }$ is denoted by $z$.
\begin{enumerate}[label=(\alph*)]
\item Determine the cube roots of $z$, giving the roots in exponential form.
The points which represent the cube roots of $z$ are denoted by $A , B$ and $C$ and these form a triangle in an Argand diagram.
\item Write down the angles that any lines of symmetry of triangle $A B C$ make with the positive real axis, justifying your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2020 Q8 [9]}}