| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus other arithmetic operations |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing basic complex number operations. Part (a) requires standard multiplication by conjugate technique, and part (b) is routine quadratic formula application with complex roots. Both are textbook exercises requiring recall of standard methods with minimal problem-solving, though slightly above average difficulty due to being Further Maths content. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR: \(\frac{8+i}{2-i} \times \frac{2+i}{2+i} = \frac{16+8i+2i-1}{4+1}\) | M1 | Multiplying top and bottom by conjugate of bottom and attempting expansion involving \(i^2 = -1\). Allow appearance of 5 in denominator with no working shown as long as clear what numerator multiplied by. |
| \(= \frac{15+10i}{5} = 3+2i\) cao | A1 | Answer must be in the requested form. Must have at least one line of working before answer (DR) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((8+i) = (2-i)(a+bi)\); \(8 = 2a+b\); \(1 = -a+2b\) | M1 | Equating to \(a+bi\) and rearranging to real and imaginary parts |
| \(\Rightarrow a=3, b=2\); \(3+2i\) cao | A1 | Answer must be in requested form. Must have at least one line of working before answer (DR) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR: \(x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4\times4\times5}}{2\times4} = \frac{8 \pm \sqrt{-16}}{8} = \frac{8 \pm 4i}{8}\) | M1 | Use of formula and finding square root of a negative number in terms of \(i\) (can be awarded even if error in calculation under square root). Condone one sign error (e.g. \(-8\) rather than \(-(-8)\) or \((-8)^2 = -64\)). Could also use completing the square. Must get as far as attempting "\(x = \ldots\)" but might make some slips. |
| \(= 1+\frac{1}{2}i\) or \(1-\frac{1}{2}i\) | A1 | or \(1 \pm \frac{1}{2}i\) or \(1 \pm 0.5i\) but must be in correct form so e.g. \(\frac{2\pm i}{2}\) is A0. Condone \(1 \pm \frac{i}{2}\) |
## Question 2:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| DR: $\frac{8+i}{2-i} \times \frac{2+i}{2+i} = \frac{16+8i+2i-1}{4+1}$ | M1 | Multiplying top and bottom by conjugate of bottom and attempting expansion involving $i^2 = -1$. Allow appearance of 5 in denominator with no working shown as long as clear what numerator multiplied by. |
| $= \frac{15+10i}{5} = 3+2i$ cao | A1 | Answer must be in the requested form. Must have at least one line of working before answer (DR) |
**Alternate method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(8+i) = (2-i)(a+bi)$; $8 = 2a+b$; $1 = -a+2b$ | M1 | Equating to $a+bi$ and rearranging to real and imaginary parts |
| $\Rightarrow a=3, b=2$; $3+2i$ cao | A1 | Answer must be in requested form. Must have at least one line of working before answer (DR) |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| DR: $x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4\times4\times5}}{2\times4} = \frac{8 \pm \sqrt{-16}}{8} = \frac{8 \pm 4i}{8}$ | M1 | Use of formula and finding square root of a negative number in terms of $i$ (can be awarded even if error in calculation under square root). Condone one sign error (e.g. $-8$ rather than $-(-8)$ or $(-8)^2 = -64$). Could also use completing the square. Must get as far as attempting "$x = \ldots$" but might make some slips. |
| $= 1+\frac{1}{2}i$ or $1-\frac{1}{2}i$ | A1 | or $1 \pm \frac{1}{2}i$ or $1 \pm 0.5i$ but must be in correct form so e.g. $\frac{2\pm i}{2}$ is **A0**. Condone $1 \pm \frac{i}{2}$ |
---2 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 8 + \mathrm { i } } { 2 - \mathrm { i } }$ in the form $\mathrm { a } + \mathrm { bi }$ where $a$ and $b$ are real.
\item Solve the equation $4 x ^ { 2 } - 8 x + 5 = 0$. Give your answer(s) in the form $\mathrm { c } + \mathrm { di }$ where $c$ and $d$ are real.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2024 Q2 [4]}}