| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Complex conjugate properties and proofs |
| Difficulty | Moderate -0.8 Part (a) is a straightforward proof requiring only the definition of complex conjugate applied to sums—pure algebraic manipulation with no insight needed. Part (b) is a routine linear equation in z once the conjugate property is applied. Both parts are below-average difficulty, being standard textbook exercises testing basic complex number definitions rather than problem-solving skills. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute for \(u\) and \(w\) and state correct conjugate of one side | B1 | |
| Express the other side without conjugates and confirm \((u+w)^* = u^* + w^*\) | B1 | Given answer. Needs explicit reference to conjugate of both sides |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute and remove conjugates to obtain a correct equation in \(x\) and \(y\) | B1 | e.g. \(x + 2 - (y+1)i + (2+i)(x+iy) = 0\) |
| Use \(i^2 = -1\) and equate real and imaginary parts to zero | M1 | |
| Obtain two correct equations in \(x\) and \(y\) | A1 | e.g. \(3x - y + 2 = 0\) and \(x + y - 1 = 0\). Allow \(xi + yi - i = 0\) |
| Solve and obtain answer \(z = -\frac{1}{4} + \frac{5}{4}i\) | A1 | Allow for real and imaginary parts stated separately |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute for $u$ and $w$ and state correct conjugate of one side | **B1** | |
| Express the other side without conjugates and confirm $(u+w)^* = u^* + w^*$ | **B1** | **Given answer.** Needs explicit reference to conjugate of both sides |
**Total: 2 marks**
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute and remove conjugates to obtain a correct equation in $x$ and $y$ | **B1** | e.g. $x + 2 - (y+1)i + (2+i)(x+iy) = 0$ |
| Use $i^2 = -1$ and equate real and imaginary parts to zero | **M1** | |
| Obtain two correct equations in $x$ and $y$ | **A1** | e.g. $3x - y + 2 = 0$ and $x + y - 1 = 0$. Allow $xi + yi - i = 0$ |
| Solve and obtain answer $z = -\frac{1}{4} + \frac{5}{4}i$ | **A1** | Allow for real and imaginary parts stated separately |
**Total: 4 marks**
---3
\begin{enumerate}[label=(\alph*)]
\item Given the complex numbers $u = a + \mathrm { i } b$ and $w = c + \mathrm { i } d$, where $a , b , c$ and $d$ are real, prove that $( u + w ) ^ { * } = u ^ { * } + w ^ { * }$.
\item Solve the equation $( z + 2 + \mathrm { i } ) ^ { * } + ( 2 + \mathrm { i } ) z = 0$, giving your answer in the form $x + \mathrm { i } y$ where $x$ and $y$ are real.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q3 [6]}}