| Exam Board | OCR |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 7 |
| Topic | Complex Numbers Arithmetic |
| Difficulty | Standard +0.8 This FP1 question requires finding complex roots via the quadratic formula, computing their product (straightforward from Vieta's formulas), then calculating the area of a triangle in the Argand diagram. While the individual steps are standard FP1 techniques, the geometric interpretation and multi-step coordination (finding coordinates, applying area formula for complex numbers) elevates this above routine exercises. The 6-mark part (ii) requires careful calculation and synthesis of complex number arithmetic with coordinate geometry. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: DR \(\left(\frac{c}{a}\right)^{17} = \frac{17}{4}\) | B1 | Could be by direct calculation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{-4 ± \sqrt{4^2 - 4 × 4 × 17}}{2 × 4}\) | M1 | Completing the square: \(4\left[\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right] + 17\) or \(\left(x + \frac{1}{2}\right)^2 = -16\) |
| Answer: \(\frac{-4 ± \sqrt{256i}}{8}\) oe | A1 | |
| Answer: \(-\frac{1}{2} ± 2i\) or \(-0.5 ± 2i\) | A1 | cao |
| Answer: Base = 4. Perp height \(\frac{1}{2} + \frac{17}{4}\) | B1ft M1ft | Seen or implied. Their \(a\omega_s - \text{Re}(\omega_1)\) provided that \(a\omega_s\) is real and \(\omega_1\) and \(\omega_s\) are a conjugate pair. or half base = 2 |
| Answer: Area \(= \frac{1}{2} × 4 × \frac{19}{4} = \frac{19}{2}\) oe | A1 |
## (i)
Answer: DR $\left(\frac{c}{a}\right)^{17} = \frac{17}{4}$ | B1 | Could be by direct calculation
## (ii)
Answer: DR Use of formula (condone one error)
$\frac{-4 ± \sqrt{4^2 - 4 × 4 × 17}}{2 × 4}$ | M1 | Completing the square: $4\left[\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right] + 17$ or $\left(x + \frac{1}{2}\right)^2 = -16$
Answer: $\frac{-4 ± \sqrt{256i}}{8}$ oe | A1 |
Answer: $-\frac{1}{2} ± 2i$ or $-0.5 ± 2i$ | A1 | cao
Answer: Base = 4. Perp height $\frac{1}{2} + \frac{17}{4}$ | B1ft M1ft | Seen or implied. Their $a\omega_s - \text{Re}(\omega_1)$ provided that $a\omega_s$ is real and $\omega_1$ and $\omega_s$ are a conjugate pair. or half base = 2
Answer: Area $= \frac{1}{2} × 4 × \frac{19}{4} = \frac{19}{2}$ oe | A1 |
---\textbf{In this question you must show detailed reasoning.}
The distinct numbers $\omega_1$ and $\omega_2$ both satisfy the quadratic equation $4x^2 + 4x + 17 = 0$.
\begin{enumerate}[label=(\roman*)]
\item Write down the value of $\omega_1 \omega_2$. [1]
\item $A$, $B$ and $C$ are the points on an Argand diagram which represent $\omega_1$, $\omega_2$ and $\omega_1 \omega_2$.
Find the area of triangle $ABC$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 AS 2017 Q4 [7]}}