| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola tangent intersection problems |
| Difficulty | Standard +0.3 This is a structured multi-part question on parabola properties with guided steps. Part (a) is standard tangent derivation using implicit differentiation, parts (b-d) involve routine coordinate geometry and recall of focus/directrix definitions. While it requires multiple techniques, each step follows logically from the previous with no novel insight needed—slightly easier than average for FP1. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{12}{7}c = -\frac{1}{t^2} \times -\frac{6}{7}c + \frac{2c}{t}\) | M1 | Substitutes \(\left(-\frac{6}{7}c,\ \frac{12}{7}c\right)\) into the equation of the tangent |
| \(\frac{12}{7}c = -\frac{1}{t^2} \times -\frac{6}{7}c + \frac{2c}{t} \Rightarrow 6t^2 - 7t - 3 = 0\) | A1 | Correct 3TQ in terms of \(t\) |
| \(6t^2 - 7t - 3 = 0 \Rightarrow (3t+1)(2t-3) = 0 \Rightarrow t =\) | M1 | Attempt to solve their 3TQ for \(t\) |
| \(t = -\frac{1}{3},\ t = \frac{3}{2} \Rightarrow \left(-\frac{1}{3}c,\ -3c\right),\ \left(\frac{3}{2}c,\ \frac{2}{3}c\right)\) | M1A1 | M1: Uses at least one of their values of \(t\) to find \(A\) or \(B\). A1: Correct coordinates |
# Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{12}{7}c = -\frac{1}{t^2} \times -\frac{6}{7}c + \frac{2c}{t}$ | M1 | Substitutes $\left(-\frac{6}{7}c,\ \frac{12}{7}c\right)$ into the equation of the tangent |
| $\frac{12}{7}c = -\frac{1}{t^2} \times -\frac{6}{7}c + \frac{2c}{t} \Rightarrow 6t^2 - 7t - 3 = 0$ | A1 | Correct 3TQ in terms of $t$ |
| $6t^2 - 7t - 3 = 0 \Rightarrow (3t+1)(2t-3) = 0 \Rightarrow t =$ | M1 | Attempt to solve their 3TQ for $t$ |
| $t = -\frac{1}{3},\ t = \frac{3}{2} \Rightarrow \left(-\frac{1}{3}c,\ -3c\right),\ \left(\frac{3}{2}c,\ \frac{2}{3}c\right)$ | M1A1 | M1: Uses at least one of their values of $t$ to find $A$ or $B$. A1: Correct coordinates |
---8. A parabola has equation $y ^ { 2 } = 4 a x , a > 0$. The point $Q \left( a q ^ { 2 } , 2 a q \right)$ lies on the parabola.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation of the tangent to the parabola at $Q$ is
$$y q = x + a q ^ { 2 }$$
This tangent meets the $y$-axis at the point $R$.
\item Find an equation of the line $l$ which passes through $R$ and is perpendicular to the tangent at $Q$.
\item Show that $l$ passes through the focus of the parabola.
\item Find the coordinates of the point where $I$ meets the directrix of the parabola.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q8}}