| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Quadratic modelling problems |
| Difficulty | Moderate -0.8 This is a straightforward quadratic modelling question requiring basic substitution (parts a,b), solving a quadratic inequality (part c), and interpretation (part d). All techniques are routine P1 level with no novel problem-solving required. Part (c) is the most demanding but still follows standard procedures for quadratic inequalities. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(P_B - P_A = 44.2 - (53 - 0.4 \times 8^2) = \ldots\) | M1 | Substitutes \(t = 0\) into equation for \(P_A\) and finds difference between 44.2 and \(P_A\) |
| awrt £16.8 million | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| £53 million | B1 | Condone 53 |
| Answer | Marks | Guidance |
|---|---|---|
| \(-1.6t + 44.2 = 53 - 0.4(t-8)^2\) | M1 | Condone errors from expanding \((t-8)^2\); also scores for use of inequality sign |
| \(0.4t^2 - 8t + \frac{84}{5} = 0 \Rightarrow t = \ldots\) | M1 | Rearranges to 3TQ = 0; attempts to solve; may use calculator |
| \(t = 10 - \sqrt{58}\) awrt 2.38 (years) | A1 | |
| \(\text{"2.38"} < t \leqslant 15\) | A1ft | From correct working; ignore upper limit if \(\geqslant 15\) |
| Answer | Marks | Guidance |
|---|---|---|
| "The share value would be negative" / "the model is known to hold for 15 years only (and 20 years is more than 15)" | B1 | Note: "\(-4.6 < 0\)" is insufficient as it doesn't refer to the model or share value of \(P\) |
# Question 5(a):
$P_B - P_A = 44.2 - (53 - 0.4 \times 8^2) = \ldots$ | M1 | Substitutes $t = 0$ into equation for $P_A$ and finds difference between 44.2 and $P_A$
awrt £16.8 million | A1 |
---
# Question 5(b):
£53 million | B1 | Condone 53
---
# Question 5(c):
$-1.6t + 44.2 = 53 - 0.4(t-8)^2$ | M1 | Condone errors from expanding $(t-8)^2$; also scores for use of inequality sign
$0.4t^2 - 8t + \frac{84}{5} = 0 \Rightarrow t = \ldots$ | M1 | Rearranges to 3TQ = 0; attempts to solve; may use calculator
$t = 10 - \sqrt{58}$ awrt 2.38 (years) | A1 |
$\text{"2.38"} < t \leqslant 15$ | A1ft | From correct working; ignore upper limit if $\geqslant 15$
---
# Question 5(d):
"The **share value** would be negative" / "the **model** is known to hold for 15 years only (and 20 years is more than 15)" | B1 | Note: "$-4.6 < 0$" is insufficient as it doesn't refer to the model or share value of $P$
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-14_563_671_255_657}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
The share value of two companies, company $A$ and company $B$, has been monitored over a 15-year period.
The share value $P _ { A }$ of company $\boldsymbol { A }$, in millions of pounds, is modelled by the equation
$$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$
where $t$ is the number of years after monitoring began.
The share value $P _ { B }$ of company $B$, in millions of pounds, is modelled by the equation
$$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$
where $t$ is the number of years after monitoring began.
Figure 2 shows a graph of both models.
Use the equations of one or both models to answer parts (a) to (d).
\begin{enumerate}[label=(\alph*)]
\item Find the difference between the share value of company $\boldsymbol { A }$ and the share value of company $\boldsymbol { B }$ at the point monitoring began.
\item State the maximum share value of company $\boldsymbol { A }$ during the 15-year period.
\item Find, using algebra and showing your working, the times during this 15-year period when the share value of company $\boldsymbol { A }$ was greater than the share value of company $\boldsymbol { B }$.
\item Explain why the model for company $\boldsymbol { A }$ should not be used to predict its share value when $t = 20$\\
\includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q5 [8]}}