| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Moderate -0.3 This is a straightforward application of modulus-argument form rules: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂). Part (b) requires finding when nθ is a multiple of π, which is routine. The question tests recall and basic manipulation rather than problem-solving, making it slightly easier than average for Further Maths Core Pure. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \( | z_1 z_2 | = 3\sqrt{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\arg(z_1 z_2) = \frac{\pi}{3} + \left(-\frac{\pi}{12}\right) = \frac{\pi}{4}\) o.e. | B1 | Deduces \(\arg(z_1 z_2) = \frac{\pi}{4}\); marks may be awarded for \(z_1 z_2 = 3\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(n = 8\) | B1ft | \(2\pi\) divided by their \(\arg(z_1 z_2)\) to give an integer; alternatively smallest positive integer multiple to make argument a multiple of \(2\pi\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \( | w^n | = (\text{their } |
| \( | w^n | = 104\,976\) |
# Question 1:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $|z_1 z_2| = 3\sqrt{2}$ | B1 | Deduces $|z_1 z_2| = 3\sqrt{2}$ |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\arg(z_1 z_2) = \frac{\pi}{3} + \left(-\frac{\pi}{12}\right) = \frac{\pi}{4}$ o.e. | B1 | Deduces $\arg(z_1 z_2) = \frac{\pi}{4}$; marks may be awarded for $z_1 z_2 = 3\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)$ |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $n = 8$ | B1ft | $2\pi$ divided by their $\arg(z_1 z_2)$ to give an integer; alternatively smallest positive integer multiple to make argument a multiple of $2\pi$ |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $|w^n| = (\text{their } |z_1 z_2|)^{\text{their } n}$ | M1 | Their answer to (a)(i) to the power of their $n$ |
| $|w^n| = 104\,976$ | A1 | cao |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$\begin{aligned}
z _ { 1 } & = 3 \left( \cos \left( \frac { \pi } { 3 } \right) + \mathrm { i } \sin \left( \frac { \pi } { 3 } \right) \right) \\
z _ { 2 } & = \sqrt { 2 } \left( \cos \left( \frac { \pi } { 12 } \right) - \mathrm { i } \sin \left( \frac { \pi } { 12 } \right) \right)
\end{aligned}$$
(a) write down the exact value of\\
(i) $\left| Z _ { 1 } Z _ { 2 } \right|$\\
(ii) $\arg \left( \mathrm { z } _ { 1 } \mathrm { z } _ { 2 } \right)$
Given that $w = z _ { 1 } z _ { 2 }$ and that $\arg \left( w ^ { n } \right) = 0$, where $n \in \mathbb { Z } ^ { + }$\\
(b) determine\\
(i) the smallest positive value of $n$\\
(ii) the corresponding value of $\left| w ^ { n } \right|$
\hfill \mbox{\textit{Edexcel CP2 2021 Q1 [5]}}