| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parameter from argument condition |
| Difficulty | Moderate -0.3 This is a straightforward multi-part FP1 question testing standard complex number operations: finding argument (routine calculator work), algebraic manipulation to express in given forms (mechanical algebra), and using an argument condition to find a parameter (requires understanding that arg = -π/2 means purely imaginary, then solving a linear equation). All parts are textbook exercises 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 forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(z + z^2 = z(1+z)\) | ||
| \(= (2 - i\sqrt{3})(1+(2-i\sqrt{3}))\) | ||
| \(= (2-i\sqrt{3})(3-i\sqrt{3})\) | ||
| \(= 6 - 2i\sqrt{3} - 3i\sqrt{3} + 3i^2\) | M1 | An attempt to multiply out the brackets to give four terms (or four terms implied). |
| \(= 6 - 2i\sqrt{3} - 3i\sqrt{3} - 3\) | M1 | Understanding that \(i^2 = -1\) and an attempt to put in the form \(a + bi\sqrt{3}\) |
| \(= 3 - 5i\sqrt{3}\) (Note: \(a=3, b=-5\)) | A1 | \(3 - 5i\sqrt{3}\) |
| [3] |
## Question 7(b) – Aliter Way 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $z + z^2 = z(1+z)$ | | |
| $= (2 - i\sqrt{3})(1+(2-i\sqrt{3}))$ | | |
| $= (2-i\sqrt{3})(3-i\sqrt{3})$ | | |
| $= 6 - 2i\sqrt{3} - 3i\sqrt{3} + 3i^2$ | M1 | An attempt to multiply out the brackets to give four terms (or four terms implied). |
| $= 6 - 2i\sqrt{3} - 3i\sqrt{3} - 3$ | M1 | Understanding that $i^2 = -1$ and an attempt to put in the form $a + bi\sqrt{3}$ |
| $= 3 - 5i\sqrt{3}$ (Note: $a=3, b=-5$) | A1 | $3 - 5i\sqrt{3}$ |
| | **[3]** | |
---7.
$$z = 2 - \mathrm { i } \sqrt { } 3$$
\begin{enumerate}[label=(\alph*)]
\item Calculate $\arg z$, giving your answer in radians to 2 decimal places.
Use algebra to express
\item $z + z ^ { 2 }$ in the form $a + b \mathrm { i } \sqrt { } 3$, where $a$ and $b$ are integers,
\item $\frac { z + 7 } { z - 1 }$ in the form $c + d \mathrm { i } \sqrt { } 3$, where $c$ and $d$ are integers.
Given that
$$w = \lambda - 3 \mathrm { i }$$
where $\lambda$ is a real constant, and $\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }$,
\item find the value of $\lambda$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2012 Q7 [11]}}