| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring understanding of complex transformations. Students must parametrise z = x + i, substitute into w = z², separate real and imaginary parts, then eliminate the parameter to find the Cartesian equation of the parabola. While methodical, it requires solid algebraic manipulation and conceptual understanding of how complex mappings work—more demanding than standard A-level but routine for FP2. |
| Spec | 4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Let \(z = x + i\) | M1 | Translates \(\text{Im}(z) = 1\) into Cartesian form |
| \(w = (x+i)^2 = (x^2 - 1) + 2xi\) | A1 | Obtains correct expression for \(w\) |
| Let \(w = u + iv\), then \(u = (x^2 - 1)\) and \(v = 2x\) | M1 | Separates real and imaginary parts, equates to \(u\) and \(v\) |
| \(\Rightarrow v^2 = 4(u+1)\), which represents a parabola | A1ft | Obtains quadratic equation and states it represents a parabola |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch of parabola with symmetry about the real axis | M1 | Parabola with symmetry about real axis |
| Accurate sketch (vertex at \(-1\) on Re axis) | A1 | Accurate sketch |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Let $z = x + i$ | M1 | Translates $\text{Im}(z) = 1$ into Cartesian form |
| $w = (x+i)^2 = (x^2 - 1) + 2xi$ | A1 | Obtains correct expression for $w$ |
| Let $w = u + iv$, then $u = (x^2 - 1)$ and $v = 2x$ | M1 | Separates real and imaginary parts, equates to $u$ and $v$ |
| $\Rightarrow v^2 = 4(u+1)$, which represents a parabola | A1ft | Obtains quadratic equation and states it represents a parabola |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch of parabola with symmetry about the real axis | M1 | Parabola with symmetry about real axis |
| Accurate sketch (vertex at $-1$ on Re axis) | A1 | Accurate sketch |
---\begin{enumerate}
\item A transformation from the $z$-plane to the $w$-plane is given by
\end{enumerate}
$$w = z ^ { 2 }$$
(a) Show that the line with equation $\operatorname { Im } ( z ) = 1$ in the $z$-plane is mapped to a parabola in the $w$-plane, giving an equation for this parabola.\\
(b) Sketch the parabola on an Argand diagram.
\hfill \mbox{\textit{Edexcel FP2 Q2 [6]}}