| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Argument relationships and tangent identities |
| Difficulty | Standard +0.3 This is a straightforward multi-part complex numbers question requiring standard techniques: plotting points on an Argand diagram, dividing complex numbers using conjugate multiplication, and using argument properties to derive a tangent identity. Part (c) requires recognizing that arg(u*/u) = -2arg(u), but this follows directly from standard argument rules. All steps are routine for P3 level with no novel insight required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show \(u\) and \(u^*\) in relatively correct positions. Must have sense of scale on axes | B1 | \(u = 3-i\), \(u^* = 3+i\). Ignore labels. |
| Show \(u^* - u\) in a relatively correct position. Must have sense of scale on axes | B1 | \(2i\). Scale only on imaginary axis is sufficient for this mark. |
| State that \(OABC\) is a parallelogram [independent of previous marks] | B1 | Ignore 'quadrilateral'. Allow 'trapezium' from correct work. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Multiply *their* numerator and the given denominator by \(3+i\) and attempt to evaluate either | M1 | Can have missing term and arithmetic errors but need \(i^2 = -1\) once, seen or implied. |
| Obtain numerator \(8 + 6i\) or denominator \(10\) | A1 | |
| State final answer \(\frac{4}{5} + \frac{3}{5}i\) or \(\frac{8}{10} + \frac{6}{10}i\) or \(0.8 + 0.6i\) | A1 | Correct answer with no working scores 0/3. |
| Alternative: Obtain two equations in \(x\) and \(y\), and attempt to solve for \(x\) or \(y\) | M1 | \(3 = 3x + y\) and \(1 = -x + 3y\) |
| Obtain \(x = \frac{4}{5}\) or \(\frac{8}{10}\) or \(0.8\), \(y = \frac{3}{5}\) or \(\frac{6}{10}\) or \(0.6\) | A1 | |
| State final answer \(\frac{4}{5} + \frac{3}{5}i\) or \(\frac{8}{10} + \frac{6}{10}i\) or \(0.8 + 0.6i\) | A1 | Correct answer with no working scores 0/3. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(\arg\frac{u^*}{u} = \arg u^* - \arg u\) or \(2\arg u^*\) | M1 | |
| Justify the given statement correctly | A1 | AG. \(\arg\frac{u^*}{u} = \tan^{-1}\frac{3}{4}\), \(\arg u^* = \tan^{-1}\frac{1}{3}\) and \(\arg u = \tan^{-1}\frac{1}{3}\) (or \(\arg u = -\tan^{-1}\frac{1}{3}\)), needed if use first expression in M1; or \(\arg\frac{u^*}{u} = \tan^{-1}\frac{3}{4}\) and \(\arg u^* = \tan^{-1}\frac{1}{3}\), needed if use second expression in M1. |
| Alternative: Use \(\tan 2A\) formula with \(\tan A = \frac{1}{3}\) | M1 | \(\tan 2A = \frac{2\tan A}{1-\tan^2 A}\), \(\tan A = \frac{1}{3}\), hence \(\tan 2A = \frac{3}{4}\) |
| Justify the given statement correctly | A1 | AG. So \(2A = \tan^{-1}\frac{3}{4} = \arg\frac{u^*}{u}\) and \(A = \tan^{-1}\frac{1}{3} = \arg u^*\), hence \(\arg\frac{u^*}{u} = 2\arg u^*\). |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show $u$ and $u^*$ in relatively correct positions. Must have sense of scale on axes | B1 | $u = 3-i$, $u^* = 3+i$. Ignore labels. |
| Show $u^* - u$ in a relatively correct position. Must have sense of scale on axes | B1 | $2i$. Scale only on imaginary axis is sufficient for this mark. |
| State that $OABC$ is a parallelogram [independent of previous marks] | B1 | Ignore 'quadrilateral'. Allow 'trapezium' from correct work. |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Multiply *their* numerator and the given denominator by $3+i$ and attempt to evaluate either | M1 | Can have missing term and arithmetic errors but need $i^2 = -1$ once, seen or implied. |
| Obtain numerator $8 + 6i$ or denominator $10$ | A1 | |
| State final answer $\frac{4}{5} + \frac{3}{5}i$ or $\frac{8}{10} + \frac{6}{10}i$ or $0.8 + 0.6i$ | A1 | Correct answer with no working scores 0/3. |
| **Alternative:** Obtain two equations in $x$ and $y$, and attempt to solve for $x$ or $y$ | M1 | $3 = 3x + y$ and $1 = -x + 3y$ |
| Obtain $x = \frac{4}{5}$ or $\frac{8}{10}$ or $0.8$, $y = \frac{3}{5}$ or $\frac{6}{10}$ or $0.6$ | A1 | |
| State final answer $\frac{4}{5} + \frac{3}{5}i$ or $\frac{8}{10} + \frac{6}{10}i$ or $0.8 + 0.6i$ | A1 | Correct answer with no working scores 0/3. |
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\arg\frac{u^*}{u} = \arg u^* - \arg u$ or $2\arg u^*$ | M1 | |
| Justify the given statement correctly | A1 | AG. $\arg\frac{u^*}{u} = \tan^{-1}\frac{3}{4}$, $\arg u^* = \tan^{-1}\frac{1}{3}$ and $\arg u = \tan^{-1}\frac{1}{3}$ (or $\arg u = -\tan^{-1}\frac{1}{3}$), needed if use first expression in M1; or $\arg\frac{u^*}{u} = \tan^{-1}\frac{3}{4}$ and $\arg u^* = \tan^{-1}\frac{1}{3}$, needed if use second expression in M1. |
| **Alternative:** Use $\tan 2A$ formula with $\tan A = \frac{1}{3}$ | M1 | $\tan 2A = \frac{2\tan A}{1-\tan^2 A}$, $\tan A = \frac{1}{3}$, hence $\tan 2A = \frac{3}{4}$ |
| Justify the given statement correctly | A1 | AG. So $2A = \tan^{-1}\frac{3}{4} = \arg\frac{u^*}{u}$ and $A = \tan^{-1}\frac{1}{3} = \arg u^*$, hence $\arg\frac{u^*}{u} = 2\arg u^*$. |5 The complex number $3 - \mathrm { i }$ is denoted by $u$.
\begin{enumerate}[label=(\alph*)]
\item Show, on an Argand diagram with origin $O$, the points $A , B$ and $C$ representing the complex numbers $u , u ^ { * }$ and $u ^ { * } - u$ respectively.
State the type of quadrilateral formed by the points $O , A , B$ and $C$.
\item Express $\frac { u ^ { * } } { u }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item By considering the argument of $\frac { u ^ { * } } { u }$, or otherwise, prove that $\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q5 [8]}}