Questions - OCR MEI C4 - A-Level Maths

OCR MEI C4 Q8

8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{070e9904-12b9-4458-b8f2-60c89b31b828-093_1013_1399_488_372} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Relative to axes $\mathrm { O } x$ (due east), $\mathrm { O } y$ (due north) and $\mathrm { O } z$ (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

Verify that the cartesian equation of the plane ABC is $3 x + 2 y + 20 z + 300 = 0$.
Find the vectors $\overrightarrow { \mathrm { DE } }$ and $\overrightarrow { \mathrm { DF } }$. Show that the vector $2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }$ is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point $\mathrm { R } ( 15,34,0 )$ in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
Write down a vector equation of the line RS. Calculate the coordinates of S.

OCR MEI C4 2005 June Q2

2 Find the first 4 terms in the binomial expansion of $\sqrt { 4 + 2 x }$. State the range of values of $x$ for which the expansion is valid.

OCR MEI C4 2005 June Q3

3 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of $\pi$.

OCR MEI C4 2005 June Q4

4 Fig. 4 shows a sketch of the region enclosed by the curve $\sqrt { 1 + \mathrm { e } ^ { - 2 x } }$, the $x$-axis, the $y$-axis and the line $x = 1$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7a1123f8-53cd-4b24-bec6-8c3bccc22653-3_517_755_1576_649} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} Find the volume of the solid generated when this region is rotated through $360 ^ { \circ }$ about the $x$-axis. Give your answer in an exact form.

OCR MEI C4 2005 June Q5

5 Solve the equation $2 \cos 2 x = 1 + \cos x$, for $0 ^ { \circ } \leqslant x < 360 ^ { \circ }$.

OCR MEI C4 2005 June Q6

6 A curve has cartesian equation $y ^ { 2 } - x ^ { 2 } = 4$.

Verify that $$x = t - \frac { 1 } { t } , \quad y = t + \frac { 1 } { t } ,$$ are parametric equations of the curve.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - 1 ) ( t + 1 ) } { t ^ { 2 } + 1 }$. Hence find the coordinates of the stationary points of the curve. Section B (36 marks)

OCR MEI C4 2005 June Q7

7 In a chemical process, the mass $M$ grams of a chemical at time $t$ minutes is modelled by the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = \frac { M } { t \left( 1 + t ^ { 2 } \right) }$$

Find $\int \frac { t } { 1 + t ^ { 2 } } \mathrm {~d} t$.
Find constants $A , B$ and $C$ such that $$\frac { 1 } { t \left( 1 + t ^ { 2 } \right) } = \frac { A } { t } + \frac { B t + C } { 1 + t ^ { 2 } } .$$
Use integration, together with your results in parts (i) and (ii), to show that $$M = \frac { K t } { \sqrt { 1 + t ^ { 2 } } } ,$$ where $K$ is a constant.
When $t = 1 , M = 25$. Calculate $K$. What is the mass of the chemical in the long term?

OCR MEI C4 2010 June Q5

Verify that $\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300
100
100 \end{array} \right)$ and find the length of the pipeline.
Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is $x + 2 y + 3 z = 320$.
Find the coordinates of the point where the pipeline meets the layer of rock.
By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer. 8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is $( \pi - 1 ) : ( \pi + 1 )$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. Find the gradient of the track at A .
Show that, when the gradient of the track is $1 , \theta$ satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
Express $\cos \theta - 4 \sin \theta$ in the form $R \cos ( \theta + \alpha )$. Hence solve the equation $\cos \theta - 4 \sin \theta = 2$ for $0 \leqslant \theta \leqslant 2 \pi$. {www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{ADVANCED GCE
MATHEMATICS (MEI)} 4754B
Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
- Insert (inserted)
- MEI Examination Formulae and Tables (MF2)
\section*{Other Materials Required:}
- Rough paper
- Scientific or graphical calculator
Wednesday 9 June 2010 Afternoon
\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
[0pt] [An answer obtained from the graph will be given no marks.]
3
In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
The text then goes on to state that the emissions per extra passenger on this journey are less than $\frac { 1 } { 2 } \mathrm {~kg}$. Justify this figure.
$\_\_\_\_$
$\_\_\_\_$
4 The daily number of trains, $n$, on a line in another country may be modelled by the function defined below, where $P$ is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 }
& n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 }
& n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 }
& n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 }
& n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 }
& \ldots \text { and so on } \ldots \end{aligned}$$
Sketch the graph of $n$ against $P$.
Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
$\_\_\_\_$

OCR MEI C4 Q6

6 Use the Insert provided for this question. The graph of $y = \tan x$ is given on the Insert.
On this graph sketch the graph of $y = \operatorname { cotx }$.
Show clearly where your graph crosses the graph of $y = \tan x$ and indicate the asymptotes.

OCR MEI C4 2006 January Q1

1 Solve the equation $\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3$.

OCR MEI C4 2006 January Q2

2 A curve is defined parametrically by the equations $$x = t - \ln t , \quad y = t + \ln t \quad ( t > 0 )$$ Find the gradient of the curve at the point where $t = 2$.

OCR MEI C4 2006 January Q3

3 A triangle ABC has vertices $\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )$ and $\mathrm { C } ( 4,8,3 )$. By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle.

OCR MEI C4 2006 January Q4

4 Solve the equation $2 \sin 2 \theta + \cos 2 \theta = 1$, for $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$.

OCR MEI C4 2006 January Q5

5

Find the cartesian equation of the plane through the point ( $2 , - 1,4$ ) with normal vector $$\mathbf { n } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right)$$
Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7
12
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
3
2 \end{array} \right)$$

OCR MEI C4 2006 January Q6

6

Find the first three non-zero terms of the binomial expansion of $\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }$ for $| x | < 2$.
Use this result to find an approximation for $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to
4 significant figures.
Given that $\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c$, evaluate $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to 4 significant figures.

OCR MEI C4 2006 January Q7

7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distance $y$ metres from the line TOA. Other distances and angles are as shown. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{897205bc-2f93-4628-8f21-2ec7fd3b3699-3_509_629_513_715} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Show that $\theta = \beta - \alpha$, and hence that $\tan \theta = \frac { 6 y } { 160 + y ^ { 2 } }$. Calculate the angle $\theta$ when $y = 6$.
By differentiating implicitly, show that $\frac { \mathrm { d } \theta } { \mathrm { d } y } = \frac { 6 \left( 160 - y ^ { 2 } \right) } { \left( 160 + y ^ { 2 } \right) ^ { 2 } } \cos ^ { 2 } \theta$.
Use this result to find the value of $y$ that maximises the angle $\theta$. Calculate this maximum value of $\theta$. [You need not verify that this value is indeed a maximum.]

OCR MEI C4 2006 January Q8

1 marks

8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population $x$, in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t } ,$$ where $t$ is the time in years, and $a$ and $k$ are constants. When $t = 0 , x = 2.5$.

Show that $\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }$.
Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate $a$ and $k$.
What is the long-term population of red squirrels predicted by this model? The population $y$, in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 } .$$ When $t = 0 , y = 1$.
Express $\frac { 1 } { 2 y - y ^ { 2 } }$ in partial fractions.
Hence show by integration that $\ln \left( \frac { y } { 2 - y } \right) = 2 t$. Show that $y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }$.
What is the long-term population of grey squirrels predicted by this model? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4)
Paper B: Comprehension
Monday
23 JANUARY 2006
Afternooon U Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2)
4754(B) \section*{Up to 1 hour
$$\text { o to } 1 \text { hour }$$} \section*{TIME} \section*{Up to 1 hour}
- Write your name, centre number and candidate number in the spaces at the top of this page.
- Answer all the questions.
- Write your answers in the spaces provided on the question paper.
- You are permitted to use a graphical calculator in this paper.
- Final answers should be given to a degree of accuracy appropriate to the context.
- The number of marks is given in brackets [ ] at the end of each question or part question.
- The insert contains the text for use with the questions.
- You may find it helpful to make notes and do some calculations as you read the passage.
- You are not required to hand in these notes with your question paper.
- You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
- The total number of marks for this section is 18.
For Examiner's Use
Qu. Mark
1
2
3
4
5
Total
1 Line 59 says "Again Party G just misses out; if there had been 7 seats G would have got the last one." Where is the evidence for this in the article? 26 parties, P, Q, R, S, T and U take part in an election for 7 seats. Their results are shown in the table below.
Party Votes (\%)
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Use the Trial-and-Improvement method, starting with values of $10 \%$ and $14 \%$, to find an acceptance percentage for 7 seats, and state the allocation of the seats.
Acceptance percentage, $\boldsymbol { a }$ \% 10\% 14\%
Party Votes (\%) Seats Seats Seats Seats Seats
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Total seats
Seat Allocation $\quad \mathrm { P } \ldots$... $\mathrm { Q } \ldots$ R ... S ... T ... $\mathrm { U } \ldots$.
Now apply the d'Hondt Formula to the same figures to find the allocation of the seats.
Round
Party 1 2 3 4 5 6 7 Residual
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Seat allocated to
Seat Allocation $\mathrm { P } \ldots$. Q … $\mathrm { R } \ldots$. S … T … $\mathrm { U } \ldots$. 3 In this question, use the figures for the example used in Table 5 in the article, the notation described in the section "Equivalence of the two methods" and the value of 11 found for $a$ in Table 4. Treating Party E as Party 5, verify that $\frac { V _ { 5 } } { N _ { 5 } + 1 } < a \leqslant \frac { V _ { 5 } } { N _ { 5 } }$.
4 Some of the intervals illustrated by the lines in the graph in Fig. 8 are given in this table.
Seats Interval Seats Interval
1 $22.2 < a \leqslant 27.0$ 5
2 $16.6 < a \leqslant 22.2$ 6 $10.6 < a \leqslant 11.1$
3 7
4
Describe briefly, giving an example, the relationship between the end-points of these intervals and the values in Table 5, which is reproduced below.

Complete the table above. \begin{table}[h]

	Round
Party	1	2	3	4	5	6	Residual
A	22.2	22.2	11.1	11.1	11.1	11.1	7.4
B	6.1	6.1	6.1	6.1	6.1	6.1	6.1
C	27.0	13.5	13.5	13.5	9.0	9.0	9.0
D	16.6	16.6	16.6	8.3	8.3	8.3	8.3
E	11.2	11.2	11.2	11.2	11.2	5.6	5.6
F	3.7	3.7	3.7	3.7	3.7	3.7	3.7
G	10.6	10.6	10.6	10.6	10.6	10.6	10.6
H	2.6	2.6	2.6	2.6	2.6	2.6	2.6
Seat allocated to	C	A	D	C	E	A

\captionsetup{labelformat=empty} \caption{Table 5}

\end{table} 5 The ends of the vertical lines in Fig. 8 are marked with circles. Those at the tops of the lines are filled in, e.g. • whereas those at the bottom are not, e.g. o.

Relate this distinction to the use of inequality signs.
Show that the inequality on line 102 can be rearranged to give $0 \leqslant V _ { k } - N _ { k } a < a$. [1]
Hence justify the use of the inequality signs in line 102.

OCR MEI C4 2006 June Q1

1 Fig. 1 shows part of the graph of $y = \sin x - \sqrt { 3 } \cos x$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-2_467_629_468_717} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Express $\sin x - \sqrt { 3 } \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi$.
Hence write down the exact coordinates of the turning point P .

OCR MEI C4 2006 June Q2

2

Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x } ,$$ where $A , B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$.
Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $( 1 + x ) ^ { - 2 }$ and $( 1 - 4 x ) ^ { - 1 }$. Hence find the first three terms of the expansion of $\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }$.

OCR MEI C4 2006 June Q3

3 Given that $\sin ( \theta + \alpha ) = 2 \sin \theta$, show that $\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }$. Hence solve the equation $\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C4 2006 June Q4

4

The number of bacteria in a colony is increasing at a rate that is proportional to the square root of the number of bacteria present. Form a differential equation relating $x$, the number of bacteria, to the time $t$.
In another colony, the number of bacteria, $y$, after time $t$ minutes is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = \frac { 10000 } { \sqrt { y } } .$$ Find $y$ in terms of $t$, given that $y = 900$ when $t = 0$. Hence find the number of bacteria after 10 minutes.

OCR MEI C4 2006 June Q5

5

Show that $\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 x } ( 1 + 2 x ) + c$. A vase is made in the shape of the volume of revolution of the curve $y = x ^ { 1 / 2 } \mathrm { e } ^ { - x }$ about the $x$-axis between $x = 0$ and $x = 2$ (see Fig. 5). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
Show that this volume of revolution is $\frac { 1 } { 4 } \pi \left( 1 - \frac { 5 } { \mathrm { e } ^ { 4 } } \right)$. Fig. 6 shows the arch ABCD of a bridge. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The section from B to C is part of the curve OBCE with parametric equations $$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ where $a$ is a constant.
Find, in terms of $a$,
(A) the length of the straight line OE,
(B) the maximum height of the arch.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. The straight line sections AB and CD are inclined at $30 ^ { \circ }$ to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the $x$-axis. BF is parallel to the $y$-axis.
Show that at the point B the parameter $\theta$ satisfies the equation $$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 - \cos \theta )$$ Verify that $\theta = \frac { 2 } { 3 } \pi$ is a solution of this equation.
Hence show that $\mathrm { BF } = \frac { 3 } { 2 } a$, and find OF in terms of $a$, giving your answer exactly.
Find BC and AF in terms of $a$. Given that the straight line distance AD is 20 metres, calculate the value of $a$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
Find the length AE .
Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30$$ Write down a vector normal to this plane.
Show that the vector $\left( \begin{array} { l } 4
3
5 \end{array} \right)$ is normal to the plane ABDE . Hence find the equation of the plane ABDE .
Find the angle between the planes ABC and ABDE . RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} Monday 12 JUNE 2006 Afternoon Up to 1 hour Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2) TIME Up to 1 hour
- Write your name, centre number and candidate number in the spaces at the top of this page.
- Answer all the questions.
- Write your answers in the spaces provided on the question paper.
- You are permitted to use a graphical calculator in this paper.
- The number of marks is given in brackets [ ] at the end of each question or part question.
- The insert contains the text for use with the questions.
- You may find it helpful to make notes and do some calculations as you read the passage.
- You are not required to hand in these notes with your question paper.
- You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
- The total number of marks for this paper is 18.
For Examiner's Use
Qu. Mark
1
2
3
4
5
6
Total
1 The marathon is 26 miles and 385 yards long ( 1 mile is 1760 yards). There are now several men who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speed for a whole marathon. How long does the athlete take?
2 According to the linear model, in which calendar year would the record for the men's mile first become negative?
3 Explain the statement in line 93 "According to this model the 2-hour marathon will never be run."
4 Explain how the equation in line 49, $$R = L + ( U - L ) \mathrm { e } ^ { - k t } ,$$ is consistent with Fig. 2
initially,
for large values of $t$.
$\_\_\_\_$
5 A model for an athletics record has the form $$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$
Sketch the graph of $R$ against $t$, showing $A$ and $B$ on your graph.
Name one event for which this might be an appropriate model.
\includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}
$\_\_\_\_$

OCR MEI C4 2006 June Q6

6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$

What is the value of $t$ for the year 2012?
What record time does this model predict for the year 2012?
$\_\_\_\_$
$\_\_\_\_$

OCR MEI C4 2008 June Q1

1 Express $\frac { x } { x ^ { 2 } - 4 } + \frac { 2 } { x + 2 }$ as a single fraction, simplifying your answer.

OCR MEI C4 2008 June Q2

2 Fig. 2 shows the curve $y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The region bounded by the curve, the $x$-axis, the $y$-axis and the line $x = 1$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Show that the volume of the solid of revolution produced is $\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)$.

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