| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2021 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | ln(y) vs ln(x) linear graph |
| Difficulty | Moderate -0.3 This is a standard logarithmic linearization question requiring taking logs of both sides (part a is a 'show that'), finding gradient and intercept from two points, then converting back to exponential form. While it involves multiple steps, each is routine and this type of question appears frequently in A-level syllabi. Slightly easier than average due to the scaffolded structure and straightforward arithmetic. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(T = al^b \Rightarrow \log_{10}T = \log_{10}a + \log_{10}l^b\) | M1 | 2.1 — Takes logs of both sides, shows addition law |
| \(\Rightarrow \log_{10}T = \log_{10}a + b\log_{10}l\) or \(\log_{10}T = b\log_{10}l + \log_{10}a\) | A1* | 1.1b — Uses power law; bases must appear in final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(b = 0.495\) or \(b = \frac{45}{91}\) | B1 | 2.2a — Deduces correct value of \(b\) |
| \(0 = \text{"0.495"}\times(-0.7) + \log_{10}a \Rightarrow a = 10^{0.346\ldots}\) or \(0.45 = \text{"0.495"}\times 0.21 + \log_{10}a \Rightarrow a = 10^{0.346\ldots}\) | M1 | 3.1a — Correct strategy to find \(a\); substitutes point with their \(b\); correct log work to find \(a\) |
| \(T = 2.22l^{0.495}\) | A1 | 3.3 — Complete equation; allow awrt 2.22 and awrt 0.495 or \(\frac{45}{91}\); must see equation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The time taken for one swing of a pendulum of length 1 m | B1 | 3.2a — Correct interpretation |
# Question 10:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $T = al^b \Rightarrow \log_{10}T = \log_{10}a + \log_{10}l^b$ | M1 | 2.1 — Takes logs of both sides, shows addition law |
| $\Rightarrow \log_{10}T = \log_{10}a + b\log_{10}l$ or $\log_{10}T = b\log_{10}l + \log_{10}a$ | A1* | 1.1b — Uses power law; bases must appear in final answer |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $b = 0.495$ or $b = \frac{45}{91}$ | B1 | 2.2a — Deduces correct value of $b$ |
| $0 = \text{"0.495"}\times(-0.7) + \log_{10}a \Rightarrow a = 10^{0.346\ldots}$ or $0.45 = \text{"0.495"}\times 0.21 + \log_{10}a \Rightarrow a = 10^{0.346\ldots}$ | M1 | 3.1a — Correct strategy to find $a$; substitutes point with their $b$; correct log work to find $a$ |
| $T = 2.22l^{0.495}$ | A1 | 3.3 — Complete equation; allow awrt 2.22 and awrt 0.495 or $\frac{45}{91}$; must see equation |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The time taken for one swing of a pendulum of length 1 m | B1 | 3.2a — Correct interpretation |
---\begin{enumerate}
\item The time, $T$ seconds, that a pendulum takes to complete one swing is modelled by the formula
\end{enumerate}
$$T = a l ^ { b }$$
where $l$ metres is the length of the pendulum and $a$ and $b$ are constants.\\
(a) Show that this relationship can be written in the form
$$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-26_581_888_749_625}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
A student carried out an experiment to find the values of the constants $a$ and $b$.\\
The student recorded the value of $T$ for different values of $l$.\\
Figure 3 shows the linear relationship between $\log _ { 10 } l$ and $\log _ { 10 } T$ for the student's data.\\
The straight line passes through the points $( - 0.7,0 )$ and $( 0.21,0.45 )$\\
Using this information,\\
(b) find a complete equation for the model in the form
$$T = a l ^ { b }$$
giving the value of $a$ and the value of $b$, each to 3 significant figures.\\
(c) With reference to the model, interpret the value of the constant $a$.\\
\hfill \mbox{\textit{Edexcel Paper 2 2021 Q10 [6]}}