| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Regression model parameter estimation |
| Difficulty | Moderate -0.8 This is a straightforward parameter estimation problem requiring substitution of two data points into the given model to form simultaneous equations, then solving for a and b. Part (b) asks for basic model evaluation by checking a prediction. The mathematics involves only simple algebra (simultaneous equations) and arithmetic—no complex statistical concepts, calculus, or novel problem-solving required. This is easier than average A-level content. |
| Spec | 1.02z Models in context: use functions in modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to use \(h^2=at+b\) with either \(t=2, h=2.6\) or \(t=10, h=5.1\) | M1 | |
| Correct equations: \(2a+b=6.76\) and \(10a+b=26.01\) | A1 | |
| Solves simultaneously to find values for \(a\) and \(b\) | dM1 | Dependent on first M1 |
| \(h^2 = 2.41t + 1.95\) (cao) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes \(t=20\) into \(h^2=2.41t+1.95\) and finds \(h\) or \(h^2\); OR substitutes \(h=7\) into \(h^2=2.41t+1.95\) and finds \(t\) | M1 | |
| Compares model with true values and concludes "good model" with minimal reason. E.g. I: finds \(h=7.08\,\text{m}\) and states close to \(7\,\text{m}\); E.g. II: finds \(t=19.5\) years and states \(19.5\approx20\) years | A1 | AO 3.5a |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to use given equation with either piece of information to form one correct equation, e.g. \(2.6^2 = 2a+b\) or \(2.6 = \sqrt{2a+b}\) | M1 | Unsimplified equations allowed |
| Two correct (and different) equations, may be unsimplified | A1 | |
| Solves simultaneously to find values for \(a\) and \(b\) | dM1 | Dependent on previous M; calculators may be used |
| Full equation of model with \(a\) and \(b\) to exactly 3sf: \(h^2 = 2.41t + 1.95\) or \(h = \sqrt{2.41t + 1.95}\) | A1 | Not scored for values of \(a\) and \(b\) alone; if they square root each term e.g. \(h = 1.55t + 1.40\) award A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes \(t=20\) into \(h^2 = 2.41t+1.95\) to find \(h\) or \(h^2\), OR substitutes \(h=7\) to find \(t\) | M1 | Equation must be correct form; \(\sqrt{}\) must be used appropriately |
| Compares \(h = 7.08\)m to \(7\)m (using \(h^2\)) or \(t = 19.5\) years to \(20\) years; makes valid conclusion with reason | A1 | Requires: statement model is "good"/"accurate"; reason e.g. "values are close"; model with \(a\) awrt \(2.4\), \(b \in [1.9, 2.0]\); correct calculations |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to use $h^2=at+b$ with either $t=2, h=2.6$ or $t=10, h=5.1$ | M1 | |
| Correct equations: $2a+b=6.76$ and $10a+b=26.01$ | A1 | |
| Solves simultaneously to find values for $a$ and $b$ | dM1 | Dependent on first M1 |
| $h^2 = 2.41t + 1.95$ (cao) | A1 | |
---
## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $t=20$ into $h^2=2.41t+1.95$ and finds $h$ or $h^2$; OR substitutes $h=7$ into $h^2=2.41t+1.95$ and finds $t$ | M1 | |
| Compares model with true values and concludes "good model" with minimal reason. E.g. I: finds $h=7.08\,\text{m}$ and states close to $7\,\text{m}$; E.g. II: finds $t=19.5$ years and states $19.5\approx20$ years | A1 | AO 3.5a |
# Question (a) [Model Equation]:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to use given equation with either piece of information to form one correct equation, e.g. $2.6^2 = 2a+b$ or $2.6 = \sqrt{2a+b}$ | M1 | Unsimplified equations allowed |
| Two correct (and different) equations, may be unsimplified | A1 | |
| Solves simultaneously to find values for $a$ and $b$ | dM1 | Dependent on previous M; calculators may be used |
| Full equation of model with $a$ and $b$ to exactly 3sf: $h^2 = 2.41t + 1.95$ or $h = \sqrt{2.41t + 1.95}$ | A1 | Not scored for values of $a$ and $b$ alone; if they square root each term e.g. $h = 1.55t + 1.40$ award A0 |
**Special case:** Using $h = at + b$: For $2.6 = 2a+b$, $5.1 = 10a+b \Rightarrow h = 0.3125t + 1.975$ or $h = 0.313t + 1.98$ scores M1 and A1 only. Maximum 1100 00.
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# Question (b) [Model Validation]:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $t=20$ into $h^2 = 2.41t+1.95$ to find $h$ or $h^2$, OR substitutes $h=7$ to find $t$ | M1 | Equation must be correct form; $\sqrt{}$ must be used appropriately |
| Compares $h = 7.08$m to $7$m (using $h^2$) or $t = 19.5$ years to $20$ years; makes valid conclusion with reason | A1 | Requires: statement model is "good"/"accurate"; reason e.g. "values are close"; model with $a$ awrt $2.4$, $b \in [1.9, 2.0]$; correct calculations |
---\begin{enumerate}
\item The height, $h$ metres, of a tree, $t$ years after being planted, is modelled by the equation
\end{enumerate}
$$h ^ { 2 } = a t + b \quad 0 \leqslant t < 25$$
where $a$ and $b$ are constants.\\
Given that
\begin{itemize}
\item the height of the tree was 2.60 m , exactly 2 years after being planted
\item the height of the tree was 5.10 m , exactly 10 years after being planted\\
(a) find a complete equation for the model, giving the values of $a$ and $b$ to 3 significant figures.
\end{itemize}
Given that the height of the tree was 7 m , exactly 20 years after being planted\\
(b) evaluate the model, giving reasons for your answer.
\hfill \mbox{\textit{Edexcel Paper 1 2022 Q5 [6]}}