| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Argument relationships and tangent identities |
| Difficulty | Standard +0.3 Part (a) is routine complex division requiring multiplication by conjugate. Part (b) is straightforward plotting. Part (c) requires understanding that arg(u) = arg(numerator) - arg(denominator) and applying this to derive a tangent identity, which is a standard P3 technique but requires some geometric insight. Overall slightly easier than average due to the guided structure. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02h Square roots: of complex numbers4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Multiply numerator and denominator by \(1+i\), or equivalent | M1 | Must multiply out |
| Obtain numerator \(6 + 8i\) or denominator \(2\) | A1 | |
| Obtain final answer \(u = 3 + 4i\) | A1 | |
| Alternative method: Multiply out \((1-i)(x+iy) = 7+i\) and compare real and imaginary parts | M1 | |
| Obtain \(x + y = 7\) or \(y - x = 1\) | A1 | |
| Obtain final answer \(u = 3 + 4i\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show the point \(A\) representing \(u\) in a relatively correct position | B1 FT | The FT is on \(xy \neq 0\). |
| Show the other two points \(B\) and \(C\) in relatively correct positions: approximately equal distance above/below real axis | B1 | Take the position of \(A\) as a guide to 'scale' if axes not marked |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\arg(1-i) = -\frac{1}{4}\pi\) | B1 | \(\text{Arg}C\) |
| Substitute exact arguments in \(\arg(7+i) - \arg(1-i) = \arg u\) | M1 | Must see a statement about the relationship between the Args e.g. \(\text{Arg}A = \text{Arg}B - \text{Arg}C\) or equivalent exact method |
| Obtain \(\tan^{-1}\left(\frac{4}{3}\right) = \tan^{-1}\left(\frac{1}{7}\right) + \frac{1}{4}\pi\) correctly | A1 | Obtain given answer correctly from *their* \(u = k(3+4i)\) |
| Total: 3 |
## Question 6:
### Part 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Multiply numerator and denominator by $1+i$, or equivalent | M1 | Must multiply out |
| Obtain numerator $6 + 8i$ or denominator $2$ | A1 | |
| Obtain final answer $u = 3 + 4i$ | A1 | |
| **Alternative method:** Multiply out $(1-i)(x+iy) = 7+i$ and compare real and imaginary parts | M1 | |
| Obtain $x + y = 7$ or $y - x = 1$ | A1 | |
| Obtain final answer $u = 3 + 4i$ | A1 | |
| | **3** | |
### Part 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show the point $A$ representing $u$ in a relatively correct position | B1 FT | The FT is on $xy \neq 0$. |
| Show the other two points $B$ and $C$ in relatively correct positions: approximately equal distance above/below real axis | B1 | Take the position of $A$ as a guide to 'scale' if axes not marked |
| | **2** | |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\arg(1-i) = -\frac{1}{4}\pi$ | B1 | $\text{Arg}C$ |
| Substitute exact arguments in $\arg(7+i) - \arg(1-i) = \arg u$ | M1 | Must see a statement about the relationship between the Args e.g. $\text{Arg}A = \text{Arg}B - \text{Arg}C$ or equivalent exact method |
| Obtain $\tan^{-1}\left(\frac{4}{3}\right) = \tan^{-1}\left(\frac{1}{7}\right) + \frac{1}{4}\pi$ correctly | A1 | Obtain given answer correctly from *their* $u = k(3+4i)$ |
| **Total: 3** | | |
---6 The complex number $u$ is defined by
$$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$
\begin{enumerate}[label=(\alph*)]
\item Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item Show on a sketch of an Argand diagram the points $A , B$ and $C$ representing $u , 7 + \mathrm { i }$ and $1 - \mathrm { i }$ respectively.
\item By considering the arguments of $7 + \mathrm { i }$ and $1 - \mathrm { i }$, show that
$$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q6 [8]}}