| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parameter from argument condition |
| Difficulty | Standard +0.3 This is a straightforward FP1 question testing standard complex number techniques: division by multiplying by conjugate, using argument condition to find a parameter, and using modulus condition to solve a quadratic. All steps are routine applications of formulas with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(w = \frac{(p-4\text{i})(2+3\text{i})}{(2-3\text{i})(2+3\text{i})}\) | M1 | Multiplies by \(\frac{(2+3\text{i})}{(2+3\text{i})}\) |
| \(= \left(\frac{2p+12}{13}\right) + \left(\frac{3p-8}{13}\right)\text{i}\) | A1 | At least one of real or imaginary part correct (expanded, may be unsimplified) |
| A1 | Correct \(w\) in simplest form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((a+\text{i}b)(2-3\text{i}) = (p-4\text{i})\) | M1 | Multiplies out to obtain 2 equations in two unknowns |
| \(2a+3b = p\), \(3a-2b = 4\) | A1 | At least one of real or imaginary part correct |
| \(= \left(\frac{2p+12}{13}\right) + \left(\frac{3p-8}{13}\right)\text{i}\) | A1 | Correct \(w\) in simplest form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\arg w = \frac{\pi}{4} \Rightarrow 2p+12 = 3p-8\) | M1 | Sets numerators of real part of \(w\) equal to imaginary part, or if arctan used, requires evidence of \(\tan\frac{\pi}{4} = 1\) |
| \(p = 20\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z = (1-\lambda\text{i})(4+3\text{i})\), \( | z | = 45\) |
| \(\sqrt{1+\lambda^2}\sqrt{4^2+3^2} = 45\) | M1 | Attempts to apply \( |
| A1 | Correct equation | |
| \(\lambda^2 = 9^2 - 1 \Rightarrow \lambda = \pm 4\sqrt{5}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z = (4+3\lambda) + (3-4\lambda)\text{i}\) | M1 | Attempt to multiply out, group real and imaginary parts and apply modulus. Also allow \((4+3\lambda)^2 + (3-4\lambda)^2\) |
| \((4+3\lambda)^2 + (3-4\lambda)^2 = 45^2\) | A1 | Correct equation |
| \(25\lambda^2 = 2000 \Rightarrow \lambda = \pm 4\sqrt{5}\) | A1 |
# Question 4:
## Part (i)(a) Way 1
| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = \frac{(p-4\text{i})(2+3\text{i})}{(2-3\text{i})(2+3\text{i})}$ | M1 | Multiplies by $\frac{(2+3\text{i})}{(2+3\text{i})}$ |
| $= \left(\frac{2p+12}{13}\right) + \left(\frac{3p-8}{13}\right)\text{i}$ | A1 | At least one of real or imaginary part correct (expanded, may be unsimplified) |
| | A1 | Correct $w$ in simplest form |
## Part (i)(a) Way 2
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(a+\text{i}b)(2-3\text{i}) = (p-4\text{i})$ | M1 | Multiplies out to obtain 2 equations in two unknowns |
| $2a+3b = p$, $3a-2b = 4$ | A1 | At least one of real or imaginary part correct |
| $= \left(\frac{2p+12}{13}\right) + \left(\frac{3p-8}{13}\right)\text{i}$ | A1 | Correct $w$ in simplest form |
## Part (i)(b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\arg w = \frac{\pi}{4} \Rightarrow 2p+12 = 3p-8$ | M1 | Sets numerators of real part of $w$ equal to imaginary part, or if arctan used, requires evidence of $\tan\frac{\pi}{4} = 1$ |
| $p = 20$ | A1 | |
## Part (ii) Way 1
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = (1-\lambda\text{i})(4+3\text{i})$, $|z| = 45$ | | |
| $\sqrt{1+\lambda^2}\sqrt{4^2+3^2} = 45$ | M1 | Attempts to apply $|(1-\lambda\text{i})(4+3\text{i})| = \sqrt{1+\lambda^2}\sqrt{4^2+3^2}$. Also allow $(1+\lambda^2)(4^2+3^2)$ |
| | A1 | Correct equation |
| $\lambda^2 = 9^2 - 1 \Rightarrow \lambda = \pm 4\sqrt{5}$ | A1 | |
## Part (ii) Way 2
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z = (4+3\lambda) + (3-4\lambda)\text{i}$ | M1 | Attempt to multiply out, group real and imaginary parts and apply modulus. Also allow $(4+3\lambda)^2 + (3-4\lambda)^2$ |
| $(4+3\lambda)^2 + (3-4\lambda)^2 = 45^2$ | A1 | Correct equation |
| $25\lambda^2 = 2000 \Rightarrow \lambda = \pm 4\sqrt{5}$ | A1 | |
---4. (i) The complex number $w$ is given by
$$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$
where $p$ is a real constant.
\begin{enumerate}[label=(\alph*)]
\item Express $w$ in the form $a + b i$, where $a$ and $b$ are real constants.
Give your answer in its simplest form in terms of $p$.
Given that $\arg w = \frac { \pi } { 4 }$
\item find the value of $p$.\\
(ii) The complex number $z$ is given by
$$z = ( 1 - \lambda i ) ( 4 + 3 i )$$
where $\lambda$ is a real constant.
Given that
$$| z | = 45$$
find the possible values of $\lambda$.\\
Give your answers as exact values in their simplest form.\\
II
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2017 Q4 [8]}}