| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Solving equations involving complex fractions |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring standard complex number operations: finding a conjugate, performing complex division by multiplying by the conjugate, and equating real/imaginary parts. While it's from FP1, the techniques are routine and mechanical with no conceptual challenges, making it slightly easier than average overall but typical for an introductory Further Pure topic. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| \(z_1^* = 2+5\text{j}\), \( | z_1 | ^2 = 4+25 = 29\) |
| \(\frac{z_1^*}{z_1} = \frac{(2+5\text{j})(2+5\text{j})}{29} = \frac{4+20\text{j}-25}{29} = \frac{-21+20\text{j}}{29}\) | M1 A1 A1 |
| Answer | Marks |
|---|---|
| \(\frac{-21}{29} = a-1 \Rightarrow a = \frac{8}{29}\), \(\frac{20}{29} = 2-b \Rightarrow b = \frac{38}{29}\) | A1 A1 |
## Question 2:
**(i)**
$z_1^* = 2+5\text{j}$, $|z_1|^2 = 4+25 = 29$ | M1 |
$\frac{z_1^*}{z_1} = \frac{(2+5\text{j})(2+5\text{j})}{29} = \frac{4+20\text{j}-25}{29} = \frac{-21+20\text{j}}{29}$ | M1 A1 A1 |
**(ii)**
$\frac{-21}{29} = a-1 \Rightarrow a = \frac{8}{29}$, $\frac{20}{29} = 2-b \Rightarrow b = \frac{38}{29}$ | A1 A1 |
---2 The complex number $z _ { 1 }$ is $2 - 5 \mathrm { j }$ and the complex number $z _ { 2 }$ is $( a - 1 ) + ( 2 - b ) \mathrm { j }$, where $a$ and $b$ are real.\\
(i) Express $\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } }$ in the form $x + y \mathrm { j }$, giving $x$ and $y$ in exact form. You must show clearly how you obtain your\\
answer.\\
(ii) Given that $\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } } = z _ { 2 }$, find the exact values of $a$ and $b$.
\hfill \mbox{\textit{OCR MEI FP1 2016 Q2 [6]}}