| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2021 |
| Session | October |
| Marks | 3 |
| 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 algebraic verification requiring substitution of parametric equations into a Cartesian equation and simplification using the common denominator (t²+1)². The algebra is routine with no conceptual difficulty or problem-solving required—students simply need to expand, substitute, and verify the identity holds. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x-3)^2 + y^2 = \left(\frac{t^2+5}{t^2+1} - 3\right)^2 + \left(\frac{4t}{t^2+1}\right)^2\) | M1 | Attempts to substitute parametric forms into Cartesian equation or lhs; there may be an (incorrect) attempt to multiply out \((x-3)^2\) term |
| \(= \frac{(2-2t^2)^2 + 16t^2}{(t^2+1)^2} = \frac{4 + 8t^2 + 4t^4}{(t^2+1)^2}\) | dM1 | Attempts to combine into single fraction using correct processing; multiplies out and collects terms on numerator |
| \(\frac{4(t^4 + 2t^2 + 1)}{(t^2+1)^2} = \frac{4(t^2+1)^2}{(t^2+1)^2} = 4^*\) | A1* | Fully correct proof showing all key steps |
# Question 13:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-3)^2 + y^2 = \left(\frac{t^2+5}{t^2+1} - 3\right)^2 + \left(\frac{4t}{t^2+1}\right)^2$ | M1 | Attempts to substitute parametric forms into Cartesian equation or lhs; there may be an (incorrect) attempt to multiply out $(x-3)^2$ term |
| $= \frac{(2-2t^2)^2 + 16t^2}{(t^2+1)^2} = \frac{4 + 8t^2 + 4t^4}{(t^2+1)^2}$ | dM1 | Attempts to combine into single fraction using correct processing; multiplies out and collects terms on numerator |
| $\frac{4(t^4 + 2t^2 + 1)}{(t^2+1)^2} = \frac{4(t^2+1)^2}{(t^2+1)^2} = 4^*$ | A1* | Fully correct proof showing all key steps |
---\begin{enumerate}
\item A curve $C$ has parametric equations
\end{enumerate}
$$x = \frac { t ^ { 2 } + 5 } { t ^ { 2 } + 1 } \quad y = \frac { 4 t } { t ^ { 2 } + 1 } \quad t \in \mathbb { R }$$
Show that all points on $C$ satisfy
$$( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$$
\hfill \mbox{\textit{Edexcel Paper 1 2021 Q13 [3]}}