| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | October |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch modulus functions involving quadratic or other non-linear |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Parts (a) and (b) involve straightforward reading of modulus function properties (vertex at x=22/3, standard intercept calculations). Part (c) is routine sketching of a parabola. Part (d) requires solving |3x-22| = (1/9)x² - 14 by cases, which is standard modulus equation technique with some algebraic manipulation, but nothing requiring novel insight—just careful execution of well-practiced methods. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02q Use intersection points: of graphs to solve equations1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(\left(\frac{22}{3}, 5\right)\) | B1 | May be listed as separate coordinates \(x=\ldots, y=\ldots\) |
| (ii) \((0, -17)\) | B1 | Correct \(y\)-intercept only; accept as coordinates or \(y=\ldots\); just \(-17\) alone is B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(5-(3x-22)=0 \Rightarrow x=\ldots\) and \(5+(3x-22)=0 \Rightarrow x=\ldots\) | M1 | Attempts to solve both equations; allow sign slips when expanding |
| \(x=9\) and \(x=\frac{17}{3}\) | A1 | Both values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct U shape symmetric about \(y\)-axis with vertex on negative \(y\)-axis | B1 | Allow tolerance with shape; curve must not clearly bend back on itself; minimum clearly on \(y\)-axis |
| Graphs meet at \((9,0)\) with \((-9,0)\) also shown | B1 | Both \(x\)-intercepts labelled or clearly stated; shape need not be correct for this mark |
| Intercept at \((0,-9)\) stated or labelled | B1 | Intercept must be on negative \(y\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Intersect at \((9,0)\) | B1 | Must be seen in (d) or clearly stated; not just marked on diagram |
| \(5+3x-22=\frac{1}{9}x^2-9\) | M1 | Sets up equation for intersection of quadratic with positive gradient line segment |
| \(\Rightarrow (x-3)(x-24)=0 \Rightarrow x=\ldots\) | dM1 | Solves equation by any valid means |
| Need smaller root \(x=3 \Rightarrow y=\ldots\) | dM1 | Depends on first M; selects correct (smaller) root and attempts \(y\) value; larger root must be rejected |
| \((3,-8)\) | A1 | \((3,-8)\) only |
# Question 7:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| (i) $\left(\frac{22}{3}, 5\right)$ | B1 | May be listed as separate coordinates $x=\ldots, y=\ldots$ |
| (ii) $(0, -17)$ | B1 | Correct $y$-intercept only; accept as coordinates or $y=\ldots$; just $-17$ alone is B0 |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5-(3x-22)=0 \Rightarrow x=\ldots$ **and** $5+(3x-22)=0 \Rightarrow x=\ldots$ | M1 | Attempts to solve both equations; allow sign slips when expanding |
| $x=9$ and $x=\frac{17}{3}$ | A1 | Both values correct |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct U shape symmetric about $y$-axis with vertex on negative $y$-axis | B1 | Allow tolerance with shape; curve must not clearly bend back on itself; minimum clearly on $y$-axis |
| Graphs meet at $(9,0)$ with $(-9,0)$ also shown | B1 | Both $x$-intercepts labelled or clearly stated; shape need not be correct for this mark |
| Intercept at $(0,-9)$ stated or labelled | B1 | Intercept must be on negative $y$-axis |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Intersect at $(9,0)$ | B1 | Must be seen in (d) or clearly stated; not just marked on diagram |
| $5+3x-22=\frac{1}{9}x^2-9$ | M1 | Sets up equation for intersection of quadratic with positive gradient line segment |
| $\Rightarrow (x-3)(x-24)=0 \Rightarrow x=\ldots$ | dM1 | Solves equation by any valid means |
| Need smaller root $x=3 \Rightarrow y=\ldots$ | dM1 | Depends on first M; selects correct (smaller) root and attempts $y$ value; larger root must be rejected |
| $(3,-8)$ | A1 | $(3,-8)$ only |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-20_624_798_219_575}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of the graph of $C _ { 1 }$ with equation
$$y = 5 - | 3 x - 22 |$$
\begin{enumerate}[label=(\alph*)]
\item Write down the coordinates of
\begin{enumerate}[label=(\roman*)]
\item the vertex of $C _ { 1 }$
\item the intersection of $C _ { 1 }$ with the $y$-axis.
\end{enumerate}\item Find the $x$ coordinates of the intersections of $C _ { 1 }$ with the $x$-axis.
Diagram 1, shown on page 21, is a copy of Figure 3.
\item On Diagram 1, sketch the curve $C _ { 2 }$ with equation
$$y = \frac { 1 } { 9 } x ^ { 2 } - 9$$
Identify clearly the coordinates of any points of intersection of $C _ { 2 }$ with the coordinate axes.
\item Find the coordinates of the points of intersection of $C _ { 1 }$ and $C _ { 2 }$ (Solutions relying entirely on calculator technology are not acceptable.)
\includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-21_629_803_1137_573}
\section*{Diagram 1}
Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q7 [12]}}