| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus modulus/argument |
| Difficulty | Moderate -0.3 This is a straightforward multi-part FP1 complex numbers question requiring standard techniques: modulus calculation, complex multiplication/division with rationalization, solving simultaneous equations, and finding argument. All parts are routine applications of basic complex number operations 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, divide |
| Answer | Marks | Guidance |
|---|---|---|
| \(z_1 + z_2 = 5 + 5i \Rightarrow | z_1 + z_2 | = \sqrt{5^2 + 5^2}\) |
| \(\sqrt{50} \ (= 5\sqrt{2})\) | A1 cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{z_1 z_3}{z_2} = \frac{(2+3i)(a+bi)}{3+2i}\) | M1 | Substitutes for \(z_1, z_2\) and \(z_3\) and multiplies by \(\frac{3-2i}{3-2i}\) |
| \((3+2i)(3-2i) = 13\) | B1 | 13 seen. |
| \(\frac{z_1 z_3}{z_2} = \frac{(12a-5b)+(5a+12b)i}{13}\) | dM1A1 | M1: Obtains a numerator with 2 real and 2 imaginary parts. A1: As stated or \(\frac{(12a-5b)}{13} + \frac{(5a+12b)}{13}i\) ONLY. |
| Answer | Marks | Guidance |
|---|---|---|
| \(12a - 5b = 17\) | M1 | Compares real and imaginary parts to obtain 2 equations which both involve \(a\) and \(b\). Condone sign errors only. |
| Answer | Marks | Guidance |
|---|---|---|
| \(60a - 25b = 85 \Rightarrow b = -1\) | dM1 | Solves as far as \(a =\) or \(b =\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = 1, b = -1\) | A1 | Both correct. Correct answers with no working award 3/3. |
| Answer | Marks | Guidance |
|---|---|---|
| \(\arg(w) = -\tan^{-1}\left(\frac{7}{17}\right)\) | M1 | Accept use of \(\pm\tan^{-1}\) or \(\pm\tan\). awrt \(\pm 0.391\) or \(\pm 5.89\) implies M1. |
| \(= \text{awrt} -0.391\) or awrt \(5.89\) | A1 |
# Question 7:
## Part (a):
$z_1 + z_2 = 5 + 5i \Rightarrow |z_1 + z_2| = \sqrt{5^2 + 5^2}$ | M1 | Adds $z_1$ and $z_2$ and correct use of Pythagoras. i under square root award M0.
$\sqrt{50} \ (= 5\sqrt{2})$ | A1 cao |
**(2 marks)**
## Part (b):
$\frac{z_1 z_3}{z_2} = \frac{(2+3i)(a+bi)}{3+2i}$ | M1 | Substitutes for $z_1, z_2$ and $z_3$ and multiplies by $\frac{3-2i}{3-2i}$
$(3+2i)(3-2i) = 13$ | B1 | 13 seen.
$\frac{z_1 z_3}{z_2} = \frac{(12a-5b)+(5a+12b)i}{13}$ | dM1A1 | M1: Obtains a numerator with 2 real and 2 imaginary parts. A1: As stated or $\frac{(12a-5b)}{13} + \frac{(5a+12b)}{13}i$ ONLY.
**(4 marks)**
## Part (c):
$12a - 5b = 17$ | M1 | Compares real and imaginary parts to obtain 2 equations which both involve $a$ and $b$. Condone sign errors only.
$5a + 12b = -7$
$60a - 25b = 85 \Rightarrow b = -1$ | dM1 | Solves as far as $a =$ or $b =$
$60a + 144b = -84$
$a = 1, b = -1$ | A1 | Both correct. Correct answers with no working award 3/3.
**(3 marks)**
## Part (d):
$\arg(w) = -\tan^{-1}\left(\frac{7}{17}\right)$ | M1 | Accept use of $\pm\tan^{-1}$ or $\pm\tan$. awrt $\pm 0.391$ or $\pm 5.89$ implies M1.
$= \text{awrt} -0.391$ or awrt $5.89$ | A1 |
**(2 marks)**
**Total: 11 marks**
---7.
$$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\left| z _ { 1 } + z _ { 2 } \right|$.
Given that $w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }$,
\item find $w$ in terms of $a$ and $b$, giving your answer in the form $x + \mathrm { i } y , \quad x , y \in \mathbb { R }$
Given also that $w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }$,
\item find the value of $a$ and the value of $b$,
\item find $\arg w$, giving your answer in radians to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q7 [11]}}