| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Quadratic model from data points |
| Difficulty | Moderate -0.8 This is a straightforward substitution problem requiring students to form two simultaneous equations from given data points and solve for p and q, then use the model to find T. The algebra is routine (linear simultaneous equations despite the cubic appearance), and part (b) is simple substitution and cube root calculation. Below average difficulty for A-level. |
| Spec | 1.02z Models in context: use functions in modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts model at least once, e.g. \(2^3 = p \times 3^2 + q\) | M1 | Either \(2^3 = p \times 3^2 + q\) or \(2.4^3 = p \times 5^2 + q\) |
| \(9p + q = 8\) and \(25p + q = 13.8(24)\) | A1 | Two correct simplified equations; may be implied by later work |
| Solves simultaneously for at least one of \(p\) or \(q\) | dM1 | Condone slips; sight of \(p = \ldots\) or \(q = \ldots\) sufficient |
| \(p = 0.364,\ q = 4.72(4)\) | A1 | Allow fractions \(p = \frac{91}{250}\), \(q = 4\frac{181}{250}\); correct answers with no working scores full marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(T\) when \(H=5\), e.g. calculates \(\sqrt{\frac{125 - \text{"q"}}{\text{"p"}}}\) | M1 | Makes \(T^2 = \frac{125-\text{"q"}}{\text{"p"}}\) and proceeds to \(T = \ldots\); can only score if \(\frac{125-\text{"q"}}{\text{"p"}} > 0\) |
| \((T =)\ 18.2\) | A1 | Allow \(T = 18.2\) years or 18 years 2.4 months |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts model at least once, e.g. $2^3 = p \times 3^2 + q$ | M1 | Either $2^3 = p \times 3^2 + q$ or $2.4^3 = p \times 5^2 + q$ |
| $9p + q = 8$ and $25p + q = 13.8(24)$ | A1 | Two correct simplified equations; may be implied by later work |
| Solves simultaneously for at least one of $p$ or $q$ | dM1 | Condone slips; sight of $p = \ldots$ or $q = \ldots$ sufficient |
| $p = 0.364,\ q = 4.72(4)$ | A1 | Allow fractions $p = \frac{91}{250}$, $q = 4\frac{181}{250}$; correct answers with no working scores full marks |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $T$ when $H=5$, e.g. calculates $\sqrt{\frac{125 - \text{"q"}}{\text{"p"}}}$ | M1 | Makes $T^2 = \frac{125-\text{"q"}}{\text{"p"}}$ and proceeds to $T = \ldots$; can only score if $\frac{125-\text{"q"}}{\text{"p"}} > 0$ |
| $(T =)\ 18.2$ | A1 | Allow $T = 18.2$ years or 18 years 2.4 months |
---\begin{enumerate}
\item A tree was planted.
\end{enumerate}
Exactly 3 years after it was planted, the height of the tree was 2 m .
Exactly 5 years after it was planted, the height of the tree was 2.4 m .
Given that the height, $H$ metres, of the tree, $t$ years after it was planted, can be modelled by the equation
$$H ^ { 3 } = p t ^ { 2 } + q$$
where $p$ and $q$ are constants,\\
(a) find, to 3 significant figures where necessary, the value of $p$ and the value of $q$.
Exactly $T$ years after the tree was planted, its height was 5 m .\\
(b) Find the value of $T$ according to the model, giving your answer to one decimal place.\\
\hfill \mbox{\textit{Edexcel P1 2021 Q2 [6]}}