| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parameter from argument condition |
| Difficulty | Standard +0.3 This is a straightforward complex number question requiring standard techniques: (a) multiply by conjugate to get Cartesian form, (b) use tan(arg) = y/x to find the parameter. Both parts are routine applications of core complex number methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Multiply numerator and denominator by \(a + 5i\) | M1 | OE |
| Use \(i^2 = -1\) | M1 | At least once |
| Obtain answer \(\dfrac{3a-10}{a^2+25} + \dfrac{2a+15}{a^2+25}i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Multiply \(x + iy\) by \(a - 5i\) and use \(i^2 = -1\) | M1 | |
| Compare real and imaginary parts | M1 | \(3 = ax + 5y,\ 2 = ay - 5x\) |
| Obtain answer \(\dfrac{3a-10}{a^2+25} + \dfrac{2a+15}{a^2+25}i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(\text{Im}(\mathbf{a}) \div \text{Re}(\mathbf{a}) = 1\) | M1 | Or \(\text{Im}(\mathbf{a}) = \text{Re}(\mathbf{a})\) or equivalent for their \(u\) |
| Obtain answer \(a = 25\) | A1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Multiply numerator and denominator by $a + 5i$ | M1 | OE |
| Use $i^2 = -1$ | M1 | At least once |
| Obtain answer $\dfrac{3a-10}{a^2+25} + \dfrac{2a+15}{a^2+25}i$ | A1 | |
**Alternative Method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Multiply $x + iy$ by $a - 5i$ and use $i^2 = -1$ | M1 | |
| Compare real and imaginary parts | M1 | $3 = ax + 5y,\ 2 = ay - 5x$ |
| Obtain answer $\dfrac{3a-10}{a^2+25} + \dfrac{2a+15}{a^2+25}i$ | A1 | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\text{Im}(\mathbf{a}) \div \text{Re}(\mathbf{a}) = 1$ | M1 | Or $\text{Im}(\mathbf{a}) = \text{Re}(\mathbf{a})$ or equivalent for their $u$ |
| Obtain answer $a = 25$ | A1 | |
---4 The complex number $u$ is defined by $u = \frac { 3 + 2 \mathrm { i } } { a - 5 \mathrm { i } }$, where $a$ is real.
\begin{enumerate}[label=(\alph*)]
\item Express $u$ in the Cartesian form $x + \mathrm { i } y$, where $x$ and $y$ are in terms of $a$.
\item Given that $\arg u = \frac { 1 } { 4 } \pi$, find the value of $a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q4 [5]}}