| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Moderate -0.8 This question tests routine application of De Moivre's theorem and basic properties of complex conjugates. Part (a) requires simplifying using standard angle formulas (cos π/2 = 0, sin π/2 = 1) and argument rules, yielding arg u = π + π/2 = 3π/2. Part (b) is direct recall that conjugation reflects in the real axis, giving arg u* = -3π/2 or π/2. Both parts are mechanical with no problem-solving required, making this easier than average. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks |
|---|---|
| 5(a) | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 7 | M1 | SOI |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | A1 | Do not accept degrees. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 5(b) | Reflection (in the) real axis | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | B1FT | FT their exact (a). Accept 2π− their exact (a). |
Question 5:
--- 5(a) ---
5(a) | 4
−
and/or
7 7 | M1 | SOI
4 1 4 1
Allow − − π or − π
7 7 7 7
Note: Many multiply top and bottom by the
conjugate, which is fine, but to score the M1
they need to state or imply the argument of a
complex number.
5
Obtain arg u = π
7 | A1 | Do not accept degrees.
2
Question | Answer | Marks | Guidance
--- 5(b) ---
5(b) | Reflection (in the) real axis | B1 | Correct non-contradictory statement.
Condone x–axis or horizontal axis.
Need ‘reflection’.
Not ‘mirror’, ‘flip’.
5
arg u* = − π
7 | B1FT | FT their exact (a). Accept 2π− their exact (a).
Accept an ‘exact’ expression in place of an exact
value.
Need to see a value or an expression. Do not
accept argu*=−arguwithout a value seen.
2\begin{enumerate}[label=(\alph*)]
\item The complex number $u$ is given by
$$u = \frac{(\cos \frac{1}{4}\pi + i \sin \frac{1}{4}\pi)^4}{\cos \frac{1}{2}\pi - i \sin \frac{1}{2}\pi}$$
Find the exact value of $\arg u$. [2]
\item The complex numbers $u$ and $u^*$ are plotted on an Argand diagram.
Describe the single geometrical transformation that maps $u$ onto $u^*$ and state the exact value of $\arg u^*$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q5 [4]}}