| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Find intersection points |
| Difficulty | Standard +0.3 This is a straightforward FP1 parametric equations question requiring standard techniques: finding coordinates from parameters, calculating gradient of PQ, finding perpendicular line through origin, converting parametric to Cartesian form (xy = 16), and solving simultaneous equations. All steps are routine with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(t = \frac{1}{4} \Rightarrow P(1,16)\), \(t=2 \Rightarrow Q(8,2)\) | B1 | Coordinates for either \(P\) or \(Q\) correctly stated |
| \(m(PQ) = \frac{2-16}{8-1} = -2\) | M1 | Finds gradient of chord \(PQ\) with \(\frac{y_2-y_1}{x_2-x_1}\) then uses \(y = -\frac{1}{m}x\). Condone incorrect sign of gradient |
| \(m(l) = \frac{1}{2}\) | ||
| \(l: y = \frac{1}{2}x\) or \(2y = x\) | A1 oe | \(y = \frac{1}{2}x\) or \(2y = x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(xy = 16\) or \(y = \frac{16}{x}\) or \(x = \frac{16}{y}\) | B1 oe | Correct Cartesian equation. Accept \(\frac{4}{y} = \frac{x}{4}\) or \(xy = 4^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Way 1: \(\frac{1}{2}x = \frac{16}{x} \Rightarrow x^2 = 32\) | M1 | Attempts to substitute their \(l\) into either their Cartesian or parametric equations of \(H\) |
| Way 2: \(\frac{4}{t} = \frac{1}{2}(4t) \Rightarrow t^2 = 2\) | ||
| Way 3: \(2y = \frac{16}{y} \Rightarrow y^2 = 8\) | ||
| \((4\sqrt{2},\ 2\sqrt{2}),\ (-4\sqrt{2},\ -2\sqrt{2})\) | A1 | At least one set of coordinates (simplified or unsimplified) or \(x = \pm 4\sqrt{2}\), \(y = \pm 2\sqrt{2}\) |
| A1 | Both sets of simplified coordinates: \(x = 4\sqrt{2},\ y = 2\sqrt{2}\) and \(x = -4\sqrt{2},\ y = -2\sqrt{2}\) |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $t = \frac{1}{4} \Rightarrow P(1,16)$, $t=2 \Rightarrow Q(8,2)$ | B1 | Coordinates for either $P$ or $Q$ correctly stated |
| $m(PQ) = \frac{2-16}{8-1} = -2$ | M1 | Finds gradient of chord $PQ$ with $\frac{y_2-y_1}{x_2-x_1}$ then uses $y = -\frac{1}{m}x$. Condone incorrect sign of gradient |
| $m(l) = \frac{1}{2}$ | | |
| $l: y = \frac{1}{2}x$ or $2y = x$ | A1 oe | $y = \frac{1}{2}x$ or $2y = x$ |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $xy = 16$ or $y = \frac{16}{x}$ or $x = \frac{16}{y}$ | B1 oe | Correct Cartesian equation. Accept $\frac{4}{y} = \frac{x}{4}$ or $xy = 4^2$ |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Way 1: $\frac{1}{2}x = \frac{16}{x} \Rightarrow x^2 = 32$ | M1 | Attempts to substitute their $l$ into either their Cartesian or parametric equations of $H$ |
| Way 2: $\frac{4}{t} = \frac{1}{2}(4t) \Rightarrow t^2 = 2$ | | |
| Way 3: $2y = \frac{16}{y} \Rightarrow y^2 = 8$ | | |
| $(4\sqrt{2},\ 2\sqrt{2}),\ (-4\sqrt{2},\ -2\sqrt{2})$ | A1 | At least one set of coordinates (simplified or unsimplified) or $x = \pm 4\sqrt{2}$, $y = \pm 2\sqrt{2}$ |
| | A1 | Both sets of simplified coordinates: $x = 4\sqrt{2},\ y = 2\sqrt{2}$ and $x = -4\sqrt{2},\ y = -2\sqrt{2}$ |
---3. The rectangular hyperbola $H$ has parametric equations
$$x = 4 t , \quad y = \frac { 4 } { t } \quad t \neq 0$$
The points $P$ and $Q$ on this hyperbola have parameters $t = \frac { 1 } { 4 }$ and $t = 2$ respectively.\\
The line $l$ passes through the origin $O$ and is perpendicular to the line $P Q$.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for $l$.
\item Find a cartesian equation for $H$.
\item Find the exact coordinates of the two points where $l$ intersects $H$. Give your answers in their simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2017 Q3 [7]}}