| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parameter from real/imaginary condition |
| Difficulty | Standard +0.3 This is a straightforward complex number manipulation problem requiring students to equate real and imaginary parts after multiplying through by the denominator. The algebra leads to a simple quadratic equation. While it involves multiple steps, each step follows standard procedures taught in P3 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 | Mark | Guidance |
| Multiply both sides by \(a + 2i\) and attempt expansion of right-hand side | \*M1 | |
| Use of \(i^2 = -1\) seen at least once (or implied) | DM1 | e.g. \(2 + 3ai = \lambda(2a+2) + \lambda i(-a+4)\) |
| Compare real and imaginary parts to obtain an equation in \(a\) only: \(\left[2 = \lambda(2a+2),\ 3a = \lambda(-a+4)\right]\) | M1 | e.g. \(\frac{3a}{2} = \frac{-a+4}{2a+2}\). Any equivalent form. |
| Obtain \(3a^2 + 4a - 4 = 0\) from correct working | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Multiply top and bottom of left-hand side by \(a - 2i\) and attempt both expansions | \*M1 | Do not need the right-hand side at this stage. |
| Use of \(i^2 = -1\) seen at least once or implied | DM1 | e.g. \(\left[\lambda(2-i) =\right] \frac{8a + i(3a^2-4)}{a^2+4}\) |
| Compare real and imaginary parts to obtain an equation in \(a\) only | M1 | e.g. \(8a = -2(3a^2 - 4)\). Any equivalent form. |
| Obtain \(3a^2 + 4a - 4 = 0\) from correct working | A1 | AG |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve given quadratic to obtain a value of \(a\) and use this to form an equation in \(\lambda\) only (based on an equation seen in *their* working in (a) or (b)) | M1 | Can be implied by relevant working seen or a correct value for \(\lambda\) seen. |
| Obtain \(a = -2,\ \lambda = -1\) or \(a = \frac{2}{3},\ \lambda = \frac{3}{5}\) | A1 | Allow \(\frac{6}{10}\) and \(0.6\). |
| Obtain second correct pair of values | A1 | |
| Total | 3 |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply both sides by $a + 2i$ and attempt expansion of right-hand side | \*M1 | |
| Use of $i^2 = -1$ seen at least once (or implied) | DM1 | e.g. $2 + 3ai = \lambda(2a+2) + \lambda i(-a+4)$ |
| Compare real and imaginary parts to obtain an equation in $a$ only: $\left[2 = \lambda(2a+2),\ 3a = \lambda(-a+4)\right]$ | M1 | e.g. $\frac{3a}{2} = \frac{-a+4}{2a+2}$. Any equivalent form. |
| Obtain $3a^2 + 4a - 4 = 0$ from correct working | A1 | AG |
**Alternative method for question 8(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply top and bottom of left-hand side by $a - 2i$ and attempt both expansions | \*M1 | Do not need the right-hand side at this stage. |
| Use of $i^2 = -1$ seen at least once or implied | DM1 | e.g. $\left[\lambda(2-i) =\right] \frac{8a + i(3a^2-4)}{a^2+4}$ |
| Compare real and imaginary parts to obtain an equation in $a$ only | M1 | e.g. $8a = -2(3a^2 - 4)$. Any equivalent form. |
| Obtain $3a^2 + 4a - 4 = 0$ from correct working | A1 | AG |
| **Total** | **4** | |
---
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve given quadratic to obtain a value of $a$ and use this to form an equation in $\lambda$ only (based on an equation seen in *their* working in (a) or (b)) | M1 | Can be implied by relevant working seen or a correct value for $\lambda$ seen. |
| Obtain $a = -2,\ \lambda = -1$ or $a = \frac{2}{3},\ \lambda = \frac{3}{5}$ | A1 | Allow $\frac{6}{10}$ and $0.6$. |
| Obtain second correct pair of values | A1 | |
| **Total** | **3** | |
---8 It is given that $\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )$, where $a$ and $\lambda$ are real constants.
\begin{enumerate}[label=(\alph*)]
\item Show that $3 a ^ { 2 } + 4 a - 4 = 0$.
\item Hence find the possible values of $a$ and the corresponding values of $\lambda$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q8 [7]}}