| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2017 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Coordinates from geometric constraints |
| Difficulty | Moderate -0.3 This is a straightforward multi-part coordinate geometry question requiring standard techniques: midpoint formula to find C, perpendicular gradient for line l, and solving simultaneous equations for intersection E. All methods are routine textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p = 13, q = 13\) | B1 B1 [2] | B1 for either \(p=13\) or \(q=13\); score within coordinate \((13,...)\) or \((...,13)\). Just 13 scores B1B0. B1 for both \(p=13\) and \(q=13\). Allow \((13,13)\) for both marks. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Gradient \(AD/AC/DC = \frac{5-(-3)}{10-7} = \left(\frac{8}{3}\right)\) | M1 | Attempt at gradient of \(AD\) or \(AC\) using coordinates for \(C\); must attempt \(\frac{\Delta y}{\Delta x}\) subtracting both numerator and denominator |
| Gradient \(DE = -\frac{3}{8}\) | M1, A1 | Attempt using \(m_2 = -\frac{1}{m_1}\); A1 for \(-\frac{3}{8}\) or equivalent |
| \((y-5) = "-\frac{3}{8}"(x-10) \Rightarrow 3x+8y=70\) | M1A1 [5] | M1 for method of line through \((10,5)\) with changed gradient; look for \((y-5)=\) changed \(m_1(x-10)\), both brackets correct. A1 for \(3x+8y=70\) or exact equivalent; accept \(\pm A(3x+8y=70)\) where \(A \in \mathbb{N}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sub \(x=7\) into \(3x+8y=70 \Rightarrow y=\frac{49}{8}\), hence \(C=\left(7,\frac{49}{8}\right)\) | M1A1 [2] | M1: substitutes \(x=7\) into their \(3x+8y=70\); A1: \(C=\left(7,\frac{49}{8}\right)\) or exact equivalent; allow when \(x\) and \(y\) written separately. Do not allow if other answers follow |
# Question 10:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p = 13, q = 13$ | B1 B1 [2] | B1 for either $p=13$ or $q=13$; score within coordinate $(13,...)$ or $(...,13)$. Just 13 scores B1B0. B1 for both $p=13$ and $q=13$. Allow $(13,13)$ for both marks. |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Gradient $AD/AC/DC = \frac{5-(-3)}{10-7} = \left(\frac{8}{3}\right)$ | M1 | Attempt at gradient of $AD$ or $AC$ using coordinates for $C$; must attempt $\frac{\Delta y}{\Delta x}$ subtracting both numerator and denominator |
| Gradient $DE = -\frac{3}{8}$ | M1, A1 | Attempt using $m_2 = -\frac{1}{m_1}$; A1 for $-\frac{3}{8}$ or equivalent |
| $(y-5) = "-\frac{3}{8}"(x-10) \Rightarrow 3x+8y=70$ | M1A1 [5] | M1 for method of line through $(10,5)$ with changed gradient; look for $(y-5)=$ changed $m_1(x-10)$, both brackets correct. A1 for $3x+8y=70$ or exact equivalent; accept $\pm A(3x+8y=70)$ where $A \in \mathbb{N}$ |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sub $x=7$ into $3x+8y=70 \Rightarrow y=\frac{49}{8}$, hence $C=\left(7,\frac{49}{8}\right)$ | M1A1 [2] | M1: substitutes $x=7$ into their $3x+8y=70$; A1: $C=\left(7,\frac{49}{8}\right)$ or exact equivalent; allow when $x$ and $y$ written separately. Do not allow if other answers follow |
---10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-24_863_929_255_511}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Diagram NOT drawn to scale
The points $A ( 7 , - 3 ) , B ( 7,20 )$ and $C ( p , q )$ form the vertices of a triangle $A B C$, as shown in Figure 4. The point $D ( 10,5 )$ is the midpoint of $A C$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $p$ and the value of $q$.
The line $l$ passes through $D$ and is perpendicular to $A C$.
\item Find an equation for $l$, in the form $a x + b y = c$, where $a$, $b$ and $c$ are integers.
Given that the line $l$ intersects $A B$ at $E$,
\item find the exact coordinates of $E$.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2017 Q10 [9]}}