| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Graphical equation solving with auxiliary line |
| Difficulty | Moderate -0.8 This question requires finding equations of line segments from coordinates (basic gradient formula), solving simultaneous linear equations, and applying a vertical stretch transformation. All techniques are routine C1/C2 skills with straightforward arithmetic. The multi-part structure and need for careful working elevate it slightly above trivial, but it remains easier than average A-level questions. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Sets \(\frac{1}{2}x = ax + 4\) where \(a < 0\) | M1 | Attempts the smaller solution; accept setting \(\frac{1}{2}x = ax + 4\) where \(a < 0\) |
| Solves \(\frac{1}{2}x = -2x + 4 \Rightarrow \frac{5}{2}x = 4 \Rightarrow x = \frac{8}{5}\) | dM1, A1 | Sets \(\frac{1}{2}x = -2x + 4\) and proceeds to \(x = ..\) by collecting terms; \(x = \frac{8}{5}\) oe, accept 1.6 |
| Sets \(\frac{1}{2}x = 5x + b\) where \(b < 0\) | M1 | Attempts to find the larger solution; accept setting \(\frac{1}{2}x = 5x + b\) where \(b < 0\) |
| Solves \(\frac{1}{2}x = 5x - 10 \Rightarrow \frac{9}{2}x = 10 \Rightarrow x = \frac{20}{9}\) | dM1, A1 | Sets \(\frac{1}{2}x = 5x - 10\) and proceeds to \(x = ..\) by collecting terms; \(x = \frac{20}{9}\), accept exact equivalents such as \(2\frac{2}{9}\) but not 2.2 or \(2.\dot{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Sketch with points \(P(-3, 6)\), \(Q(0, 6)\), \(R(2, 0)\), \(S(4, 15)\) — any two points correct | B1 | Any two points correct either in text or on sketch; accept 6 and 2 written on correct axes |
| Same shape with all four points correct | B1 | Shape + all four points correct. Candidates may adapt the given diagram. If coordinates given on diagram and in body of script, diagram takes precedence |
## Question 14:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| Sets $\frac{1}{2}x = ax + 4$ where $a < 0$ | M1 | Attempts the smaller solution; accept setting $\frac{1}{2}x = ax + 4$ where $a < 0$ |
| Solves $\frac{1}{2}x = -2x + 4 \Rightarrow \frac{5}{2}x = 4 \Rightarrow x = \frac{8}{5}$ | dM1, A1 | Sets $\frac{1}{2}x = -2x + 4$ and proceeds to $x = ..$ by collecting terms; $x = \frac{8}{5}$ oe, accept 1.6 |
| Sets $\frac{1}{2}x = 5x + b$ where $b < 0$ | M1 | Attempts to find the larger solution; accept setting $\frac{1}{2}x = 5x + b$ where $b < 0$ |
| Solves $\frac{1}{2}x = 5x - 10 \Rightarrow \frac{9}{2}x = 10 \Rightarrow x = \frac{20}{9}$ | dM1, A1 | Sets $\frac{1}{2}x = 5x - 10$ and proceeds to $x = ..$ by collecting terms; $x = \frac{20}{9}$, accept exact equivalents such as $2\frac{2}{9}$ but not 2.2 or $2.\dot{2}$ |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Sketch with points $P(-3, 6)$, $Q(0, 6)$, $R(2, 0)$, $S(4, 15)$ — any two points correct | B1 | Any two points correct either in text or on sketch; accept 6 and 2 written on correct axes |
| Same shape with all four points correct | B1 | Shape + all four points correct. Candidates may adapt the given diagram. If coordinates given on diagram and in body of script, diagram takes precedence |
---14.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-21_831_919_127_509}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a sketch of the graph of $y = g ( x ) , - 3 \leqslant x \leqslant 4$ and part of the line $l$ with equation $y = \frac { 1 } { 2 } x$
The graph of $y = \mathrm { g } ( x )$ consists of three line segments, from $P ( - 3,4 )$ to $Q ( 0,4 )$, from $Q ( 0,4 )$ to $R ( 2,0 )$ and from $R ( 2,0 )$ to $S ( 4,10 )$.
The line $l$ intersects $y = \mathrm { g } ( x )$ at the points $A$ and $B$ as shown in Figure 4.
\begin{enumerate}[label=(\alph*)]
\item Use algebra to find the $x$ coordinate of the point $A$ and the $x$ coordinate of the point $B$.
Show each step of your working and give your answers as exact fractions.
\item Sketch the graph with equation
$$y = \frac { 3 } { 2 } g ( x ) , \quad - 3 \leqslant x \leqslant 4$$
On your sketch show the coordinates of the points to which $P , Q , R$ and $S$ are transformed.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q14 [8]}}