| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.3 This is a standard P3 linearisation question requiring students to convert y=ab^t to log₁₀P = log₁₀a + t·log₁₀b, identify the intercept and gradient, then solve for constants. The steps are routine and well-practiced in the syllabus, though it requires careful handling of logarithms. Part (c) is straightforward substitution. Slightly easier than average due to being a textbook application with no conceptual surprises. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States \(\log a = 0.68\) or \(\log b = 0.09\) | M1 | Either states \(\log a = 0.68\), \(a=10^{0.68}\), \(a=\) awrt 4.8; or any of \(\log b=0.09\), \(b=10^{0.09}\), \(b=\) awrt 1.2 |
| \(a = 4.79\) or \(b = 1.23\) | A1 | Achieves either \(a=\) awrt 4.79 or \(b=\) awrt 1.23 |
| States \(\log a = 0.68\) and \(\log b = 0.09\) | M1 | States correct equation for both \(a\) and \(b\) |
| \(a = 4.79\) and \(b = 1.23\) CSO | A1 | Achieves \(a=4.79\) and \(b=1.23\) with no incorrect work |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The percentage of the population with access to the internet at the start of 2005 | B1 | Must include "percentage", "internet", and "2005". Minimal answer: "the percentage with access to the internet in 2005". Also allow "the initial percentage with internet access" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P = 4.79\times 1.23^{10}=\) awrt 38 | M1, A1 | M1 for attempting \(4.79\times1.23^{10}\) following through on their 4.79 and 1.23. Condone attempt at \(4.79\times1.23^{11}\). A1: AWRT 38, ISW after sight of awrt 38 |
# Question 8:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States $\log a = 0.68$ or $\log b = 0.09$ | M1 | Either states $\log a = 0.68$, $a=10^{0.68}$, $a=$ awrt 4.8; or any of $\log b=0.09$, $b=10^{0.09}$, $b=$ awrt 1.2 |
| $a = 4.79$ **or** $b = 1.23$ | A1 | Achieves either $a=$ awrt 4.79 **or** $b=$ awrt 1.23 |
| States $\log a = 0.68$ **and** $\log b = 0.09$ | M1 | States correct equation for both $a$ and $b$ |
| $a = 4.79$ **and** $b = 1.23$ CSO | A1 | Achieves $a=4.79$ **and** $b=1.23$ with no incorrect work |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The percentage of the population with access to the internet at the start of 2005 | B1 | Must include "percentage", "internet", and "2005". Minimal answer: "the percentage with access to the internet in 2005". Also allow "the initial percentage with internet access" |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P = 4.79\times 1.23^{10}=$ awrt 38 | M1, A1 | M1 for attempting $4.79\times1.23^{10}$ following through on their 4.79 and 1.23. Condone attempt at $4.79\times1.23^{11}$. A1: AWRT 38, ISW after sight of awrt 38 |
---\begin{enumerate}
\item The percentage, $P$, of the population of a small country who have access to the internet, is modelled by the equation
\end{enumerate}
$$P = a b ^ { t }$$
where $a$ and $b$ are constants and $t$ is the number of years after the start of 2005\\
Using the data for the years between the start of 2005 and the start of 2010, a graph is plotted of $\log _ { 10 } P$ against $t$.
The points are found to lie approximately on a straight line with gradient 0.09 and intercept 0.68 on the $\log _ { 10 } P$ axis.\\
(a) Find, according to the model, the value of $a$ and the value of $b$, giving your answers to 2 decimal places.\\
(b) In the context of the model, give a practical interpretation of the constant $a$.\\
(c) Use the model to estimate the percentage of the population who had access to the internet at the start of 2015
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\hfill \mbox{\textit{Edexcel P3 2021 Q8 [7]}}