| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2021 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (polynomial/rational) |
| Difficulty | Standard +0.3 This is a straightforward parametric-to-Cartesian conversion requiring algebraic manipulation to eliminate the parameter t. The process involves rearranging x = 1/t + 2 to find t, substituting into y, and simplifying the rational expression. Part (b) requires finding the range from the Cartesian form, which is a standard technique. While it involves several algebraic steps, it follows a well-practiced procedure 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 |
| \(k = 2\) or \(x > 2\) | B1 | Must be seen in part (a) |
| \(t = \frac{1}{x-2} \Rightarrow y = \frac{1-\frac{2}{x-2}}{3+\frac{1}{x-2}}\) | M1 A1 | Attempts to find \(t\) in terms of \(x\) and substitutes into \(y\) |
| \(\frac{1-\frac{2}{x-2}}{3+\frac{1}{x-2}} = \frac{x-2-2}{\ldots}\) or \(\frac{\ldots}{3(x-2)+1}\) | A1 (M1 on EPEN) | Correct numerator or denominator with fraction removed (allow unsimplified) |
| \(y = \frac{x-4}{3x-5}\) | A1 | Must be \(y = \ldots\) or \(g(x) = \ldots\) present at some point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(-2 < g < \frac{1}{3}\) | M1 A1 | M1 for obtaining one of the 2 boundaries; A1 correct range. Allow \(-2 < g(x) < \frac{1}{3}\), \(-2 < y < \frac{1}{3}\), \(\left(-2, \frac{1}{3}\right)\), \(g > -2\) and \(g < \frac{1}{3}\) |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k = 2$ or $x > 2$ | B1 | Must be seen in part (a) |
| $t = \frac{1}{x-2} \Rightarrow y = \frac{1-\frac{2}{x-2}}{3+\frac{1}{x-2}}$ | M1 A1 | Attempts to find $t$ in terms of $x$ and substitutes into $y$ |
| $\frac{1-\frac{2}{x-2}}{3+\frac{1}{x-2}} = \frac{x-2-2}{\ldots}$ or $\frac{\ldots}{3(x-2)+1}$ | A1 (M1 on EPEN) | Correct numerator or denominator with fraction removed (allow unsimplified) |
| $y = \frac{x-4}{3x-5}$ | A1 | Must be $y = \ldots$ or $g(x) = \ldots$ present at some point |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $-2 < g < \frac{1}{3}$ | M1 A1 | M1 for obtaining one of the 2 boundaries; A1 correct range. Allow $-2 < g(x) < \frac{1}{3}$, $-2 < y < \frac{1}{3}$, $\left(-2, \frac{1}{3}\right)$, $g > -2$ **and** $g < \frac{1}{3}$ |
---4. The curve $C$ is defined by the parametric equations
$$x = \frac { 1 } { t } + 2 \quad y = \frac { 1 - 2 t } { 3 + t } \quad t > 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of $C$ can be written in the form $y = \mathrm { g } ( x )$ where g is the function
$$\mathrm { g } ( x ) = \frac { a x + b } { c x + d } \quad x > k$$
where $a , b , c , d$ and $k$ are integers to be found.
\item Hence, or otherwise, state the range of g .
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\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2021 Q4 [7]}}