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Questions: Trockers A-Level Pure Mathematicsugust 2020 Question Paper
This is the A-Level Pure Mathematics marking scheme from August 2020. It shows how marks are awarded, so you can mark your own answers to the matching question paper. See the question paper.
Upper $6^{\text {th }}$
LOCKDOWN PURE MATHS TEST 18
Monday 10 August 2020
ANSWERS
20 MARKS
NB: If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to $\mathbf{2}$ significant figures.
1. An Hp laptop dealer values an Hp Pavilion laptop at $$ x$ on January $1^{\text {st }}$ 2016. The value, $$ m$, of the laptop $t$ years after this date may be modelled as a continuous variable. The rate of increase of $m$ may be assumed to be directly proportional to $m^{2}$.
a) Write down a differential equation satisfied by $m$.
[2]
The rate of increase of $m$ may be assumed to be directly proportional to $m^{2}$ :
\[
\begin{array}{l}
\frac{d m}{d t} \propto m^{2}
\Rightarrow \frac{d m}{d t}=k m^{2}
\end{array}
\]
b) Show that $\frac{1}{k}\left(\frac{m-x}{m x}\right)=t$, where $k$ is a constant.
[4]
\[
\begin{array}{l}
\frac{d m}{d t}=k m^{2}
\Rightarrow \int \frac{1}{m^{2}} d m=\int k d t
\Rightarrow \int m^{-2} d m=k t+c
\Rightarrow \frac{m^{-2+1}}{-2+1}=k t+c
\Rightarrow-\frac{1}{m}=k t+c
\end{array}
\]
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