7 A drug is administered by an intravenous drip. The concentration, $x$, of the drug in the blood is measured as a fraction of its maximum level. The drug concentration after $t$ hours is modelled by the differential equation

$$\frac { \mathrm { d } x } { \mathrm {~d} t } = k \left( 1 + x - 2 x ^ { 2 } \right) ,$$

where $0 \leqslant x < 1$, and $k$ is a positive constant. Initially, $x = 0$.\\
(i) Express $\frac { 1 } { ( 1 + 2 x ) ( 1 - x ) }$ in partial fractions.\\
(ii) Hence solve the differential equation to show that $\frac { 1 + 2 x } { 1 - x } = \mathrm { e } ^ { 3 k t }$.\\
(iii) After 1 hour the drug concentration reaches $75 \%$ of its maximum value and so $x = 0.75$.

Find the value of $k$, and the time taken for the drug concentration to reach $90 \%$ of its maximum value.\\
(iv) Rearrange the equation in part (ii) to show that $x = \frac { 1 - \mathrm { e } ^ { - 3 k t } } { 1 + 2 \mathrm { e } ^ { - 3 k t } }$.

Verify that in the long term the drug concentration approaches its maximum value.

\section*{END OF QUESTION PAPER}
\section*{Tuesday 16 J une 2015 - Afternoon}
\section*{A2 GCE MATHEMATICS (MEI)}
4754/01B Applications of Advanced Mathematics (C4) Paper B: Comprehension

\section*{QUESTION PAPER}
\section*{Candidates answer on the Question Paper.}
\section*{OCR supplied materials:}
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  \item Insert (inserted)
  \item MEI Examination Formulae and Tables (MF2)
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\section*{Other materials required:}
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  \item Scientific or graphical calculator
  \item Rough paper
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Duration: Up to 1 hour\\
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PLEASE DO NOT WRITE IN THIS SPACE

2 In line 79 it says "For most journeys, more than half the journey time is composed of load time and transfer time". For what percentage of the journey time for the round trip made by car A in Table 4 is the car stationary?

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3 Using the expression on line 51, work out the answer to the question on lines 39 and 40 for the case where there are 10 upper floors and 7 people. Give your answer to 2 decimal places.

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4 In lines 89 and 90 it says "... on average there will be approximately 8 stops per trip. A round trip with 8 stops would take between 188 and 200 seconds". Explain how the figure of 188 seconds has been derived.

5 (i) Referring to Strategy 3 and lines 99 to 101, complete the table below for car C .\\
(ii) Calculate the time car C will take to transport all the people who work on floors 7 and 8 , and return to the ground floor.

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68 people make independent visits to any one of the upper floors of a building with 10 upper floors. What is the probability that at least one of the visitors goes to the top floor?

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7 On lines 94 and 95 it says "Table 4 gives the timings for round trips in which the cars are required to stop at every floor they serve; Table 2 suggests this is a common occurrence in this case". Explain how Table 2 is used to make this claim.\\
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END OF QUESTION PAPER

\hfill \mbox{\textit{OCR MEI C4 2015 Q7 [18]}}