| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting quadratic curve |
| Difficulty | Standard +0.3 This is a standard C1 question on line-curve intersection with differentiation for tangents. Part (a) requires solving a quadratic equation, parts (b-c) use basic differentiation to find tangent equations, and part (d) solves simultaneous linear equations. All techniques are routine for C1 with no novel insight required, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02q Use intersection points: of graphs to solve equations1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(x^2 - 3x + 5 = 2x + 1\) | ||
| \(x^2 - 5x + 4 = 0\) | M1 | |
| \((x - 1)(x - 4) = 0\) | M1 | |
| \(x = 1, 4\) | A1 | |
| when \(x = 1\), \(y = 2(1) + 1 = 3\) ⇒ \(P(1, 3), Q(4, 9)\) | A1 | |
| (b) \(\frac{dy}{dx} = 2x - 3\) | M1 | |
| grad \(= -1\) | A1 | |
| ⇒ \(y - 3 = -(x - 1)\) [\(y = 4 - x\)] | M1 A1 | |
| (c) grad \(= 5\) | ||
| ⇒ \(y - 9 = 5(x - 4)\) | M1 | |
| \(y - 9 = 5x - 20\) | ||
| \(y = 5x - 11\) | A1 | |
| (d) \(4 - x = 5x - 11\) | M1 | |
| \(x = \frac{5}{2}\) | A1 | |
| ⇒ \(\left(\frac{5}{2}, \frac{3}{2}\right)\) | A1 | (13 marks) |
**(a)** $x^2 - 3x + 5 = 2x + 1$ | |
$x^2 - 5x + 4 = 0$ | M1 |
$(x - 1)(x - 4) = 0$ | M1 |
$x = 1, 4$ | A1 |
when $x = 1$, $y = 2(1) + 1 = 3$ ⇒ $P(1, 3), Q(4, 9)$ | A1 |
**(b)** $\frac{dy}{dx} = 2x - 3$ | M1 |
grad $= -1$ | A1 |
⇒ $y - 3 = -(x - 1)$ [$y = 4 - x$] | M1 A1 |
**(c)** grad $= 5$ | |
⇒ $y - 9 = 5(x - 4)$ | M1 |
$y - 9 = 5x - 20$ | |
$y = 5x - 11$ | A1 |
**(d)** $4 - x = 5x - 11$ | M1 |
$x = \frac{5}{2}$ | A1 |
⇒ $\left(\frac{5}{2}, \frac{3}{2}\right)$ | A1 | (13 marks)
---
**Total: 75 marks**10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ddc2483c-fc21-4d6f-9e5b-7c48339dbc88-4_647_775_879_475}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve $y = x ^ { 2 } - 3 x + 5$ and the straight line $y = 2 x + 1$. The curve and line intersect at the points $P$ and $Q$.
\begin{enumerate}[label=(\alph*)]
\item Using algebra, show that $P$ has coordinates $( 1,3 )$ and find the coordinates of $Q$.
\item Find an equation for the tangent to the curve at $P$.
\item Show that the tangent to the curve at $Q$ has the equation $y = 5 x - 11$.
\item Find the coordinates of the point where the tangent to the curve at $P$ intersects the tangent to the curve at $Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q10 [13]}}