| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Graph y=a|bx+c|+d given: solve equation or inequality |
| Difficulty | Moderate -0.3 This is a straightforward modulus function question testing standard techniques: finding vertex coordinates, range, function composition, solving modulus inequalities, and transformations. All parts follow routine procedures taught in P3 with no novel problem-solving required. Part (c) requires splitting into cases but this is a standard textbook exercise. Part (d) on transformations is mechanical application of rules. Slightly easier than average due to the step-by-step scaffolding across multiple parts. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(\frac{13}{3}, 5\right)\) | B1 B1 | One correct coordinate; both coordinates correct; condone missing brackets |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) \geqslant 5\) | B1ft | Follow through on \(y\)-coordinate from (a); allow \(y\geqslant 5\), \(y\in[5,\infty)\), \(f\geqslant 5\); do not allow "range \(\geqslant 5\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(10\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to solve \(16-2x \ldots 3x-13+5\) or \(16-2x \ldots -3x+13+5\) | M1 | Ignore direction of inequality; allow slips in rearrangement |
| Both critical values \(2,\ \frac{24}{5}\) | A1 | Correct critical values; may be part of incorrect inequality |
| Selects inside region for critical values | dM1 | Selects inside region or both correct inequalities seen |
| \(2 < x < \frac{24}{5}\) | A1 | Allow \(x\in\left(2,\frac{24}{5}\right)\) or \(\frac{24}{5}>x>2\); must be on one line; accept \(2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a=4,\ b=\frac{1}{3}\) | B1ft B1 | One correct value ft on part (a); both correct; may be embedded within \(y=af(x+b)\) |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\frac{13}{3}, 5\right)$ | B1 B1 | One correct coordinate; both coordinates correct; condone missing brackets |
## Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) \geqslant 5$ | B1ft | Follow through on $y$-coordinate from (a); allow $y\geqslant 5$, $y\in[5,\infty)$, $f\geqslant 5$; do not allow "range $\geqslant 5$" |
## Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $10$ | B1 | |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to solve $16-2x \ldots 3x-13+5$ or $16-2x \ldots -3x+13+5$ | M1 | Ignore direction of inequality; allow slips in rearrangement |
| Both critical values $2,\ \frac{24}{5}$ | A1 | Correct critical values; may be part of incorrect inequality |
| Selects inside region for critical values | dM1 | Selects inside region or both correct inequalities seen |
| $2 < x < \frac{24}{5}$ | A1 | Allow $x\in\left(2,\frac{24}{5}\right)$ or $\frac{24}{5}>x>2$; must be on one line; accept $2<x<\frac{48}{10}$ |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a=4,\ b=\frac{1}{3}$ | B1ft B1 | One correct value ft on part (a); both correct; may be embedded within $y=af(x+b)$ |2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-06_570_604_255_673}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the graph with equation $y = \mathrm { f } ( x )$, where
$$f ( x ) = | 3 x - 13 | + 5 \quad x \in \mathbb { R }$$
The vertex of the graph is at point $P$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item State the coordinates of $P$.
\item \begin{enumerate}[label=(\roman*)]
\item State the range of f .
\item Find the value of ff(4)
\end{enumerate}\item Solve, using algebra and showing your working,
$$16 - 2 x > | 3 x - 13 | + 5$$
The graph with equation $y = \mathrm { f } ( x )$ is transformed onto the graph with equation $y = a \mathrm { f } ( x + b )$ The vertex of the graph with equation $y = a \mathrm { f } ( x + b )$ is $( 4,20 )$
Given that $a$ and $b$ are constants,
\item find the value of $a$ and the value of $b$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2021 Q2 [10]}}