| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| 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 FP1 complex numbers question requiring routine algebraic manipulation (expanding z²+2z), using the condition that imaginary part equals zero, then calculating modulus/argument with standard formulas. Part (e) requires some geometric insight but the overall question is more accessible than average A-level, being a standard textbook-style exercise with clear scaffolding across multiple parts. |
| 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, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z^2 = (a+2i)(a+2i) = (a^2-4)+4ai\) | M1 | Squares \(z\) to produce at least 3 terms |
| \(z^2+2z = (a^2-4+2a)+i(4a+4)\), so \(x=(a^2+2a-4)\) and \(y=4a+4\) | M1, A1, A1 | M1: Adds \(2z\) to \(z^2\); A1: Correct \(x\); A1: Correct \(y\) (accept \(4ai+4i\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4a+4=0 \rightarrow a=-1\) | B1 | Completely accurate cao |
| ALT: Substitute \(a=-1\) and show \(y=0\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \( | z | = \sqrt{5}\) or awrt 2.24 |
| \(\arctan(-2) = 2.03\) | M1, A1 cao | M1: use tan or arctan; A1: 2.03 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Plot \(P(-1,2)\), \(Q(-3,-4)\), \(R(-5,0)\) on Argand diagram | M1, A1, B1ft | M1: \(OP\) in correct quadrant or \(OQ\) in correct quadrant; A1: Both \(OP\) and \(OQ\) correct (2nd and 3rd quadrants); B1ft: \(OR\) on real axis to left of origin |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(OP\) and \(QR\) are parallel, and \(QR\) is twice the length of \(OP\) | B1, B1 | Or: Enlargement SF 2 (centre \(O\)) followed by translation \(\begin{pmatrix}-3\\-4\end{pmatrix}\); or Enlargement SF 2 centre \((3,4)\) or centre \(3+4i\); \(\overrightarrow{QR}=2\overrightarrow{OP}\) with clear vectors → B1B1; without vectors → B0B1 |
# Question 7:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z^2 = (a+2i)(a+2i) = (a^2-4)+4ai$ | M1 | Squares $z$ to produce at least 3 terms |
| $z^2+2z = (a^2-4+2a)+i(4a+4)$, so $x=(a^2+2a-4)$ and $y=4a+4$ | M1, A1, A1 | M1: Adds $2z$ to $z^2$; A1: Correct $x$; A1: Correct $y$ (accept $4ai+4i$) |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4a+4=0 \rightarrow a=-1$ | B1 | Completely accurate cao |
| ALT: Substitute $a=-1$ and show $y=0$ | B1 | |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $|z| = \sqrt{5}$ or awrt 2.24 | B1 | $\sqrt{5}$ or 2.24 or awrt 2.24 |
| $\arctan(-2) = 2.03$ | M1, A1 cao | M1: use tan or arctan; A1: 2.03 |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Plot $P(-1,2)$, $Q(-3,-4)$, $R(-5,0)$ on Argand diagram | M1, A1, B1ft | M1: $OP$ in correct quadrant or $OQ$ in correct quadrant; A1: Both $OP$ and $OQ$ correct (2nd and 3rd quadrants); B1ft: $OR$ on real axis to left of origin |
## Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $OP$ and $QR$ are parallel, and $QR$ is twice the length of $OP$ | B1, B1 | Or: Enlargement SF 2 (centre $O$) followed by translation $\begin{pmatrix}-3\\-4\end{pmatrix}$; or Enlargement SF 2 centre $(3,4)$ or centre $3+4i$; $\overrightarrow{QR}=2\overrightarrow{OP}$ with clear vectors → B1B1; without vectors → B0B1 |
---7. A complex number $z$ is given by
$$z = a + 2 i$$
where $a$ is a non-zero real number.
\begin{enumerate}[label=(\alph*)]
\item Find $z ^ { 2 } + 2 z$ in the form $x +$ iy where $x$ and $y$ are real expressions in terms of $a$.
Given that $z ^ { 2 } + 2 z$ is real,
\item find the value of $a$.
Using this value for $a$,
\item find the values of the modulus and argument of $z$, giving the argument in radians, and giving your answers to 3 significant figures.
\item Show the points $P , Q$ and $R$, representing the complex numbers $z , z ^ { 2 }$ and $z ^ { 2 } + 2 z$ respectively, on a single Argand diagram with origin $O$.
\item Describe fully the geometrical relationship between the line segments $O P$ and $Q R$.\\
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2016 Q7 [13]}}