OCR MEI C4 (Core Mathematics 4) 2010 June

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

2 Fig. 2 shows the curve \(y = \sqrt { 1 + x ^ { 2 } }\). \begin{figure}[h] \end{figure}

1. The following table gives some values of \(x\) and \(y\).

   \(x\)	0	0.25	0.5	0.75	1
   \(y\)	1	1.0308		1.25	1.4142

   Find the missing value of \(y\), giving your answer correct to 4 decimal places.
   Hence show that, using the trapezium rule with four strips, the shaded area is approximately 1.151 square units.
2. Jenny uses a trapezium rule with 8 strips, and obtains a value of 1.158 square units. Explain why she must have made a mistake.
3. The shaded area is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid of revolution formed.

3 The parametric equations of a curve are $$x = \cos 2 \theta , \quad y = \sin \theta \cos \theta \quad \text { for } 0 \leqslant \theta < \pi$$ Show that the cartesian equation of the curve is \(x ^ { 2 } + 4 y ^ { 2 } = 1\).
Sketch the curve.

4 Find the first three terms in the binomial expansion of \(\sqrt { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.
show that \(\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }\), where \(A\) is a constant.
(ii) Hence, given that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$ 6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)
7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300
   100
   100 \end{array} \right)\) and find the length of the pipeline.
2. 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 \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. 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}
5. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
7. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
8. 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
9. 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.
10. 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.
11. \(\_\_\_\_\)
12. \(\_\_\_\_\)
    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. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 }
    & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 }
    & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 }
    & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 }
    & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 }
    & \ldots \text { and so on } \ldots \end{aligned}$$
13. Sketch the graph of \(n\) against \(P\).
14. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
15. 
    \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
16. \(\_\_\_\_\)

6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)

7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300
   100
   100 \end{array} \right)\) and find the length of the pipeline.
2. 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 \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. 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] \end{figure}

1. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
3. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
4. 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
5. 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.
6. 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.
7. \(\_\_\_\_\)
8. \(\_\_\_\_\)
   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. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 }
   & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 }
   & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 }
   & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 }
   & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 }
   & \ldots \text { and so on } \ldots \end{aligned}$$
9. Sketch the graph of \(n\) against \(P\).
10. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
11. 
    \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
12. \(\_\_\_\_\)
    5 The FGW website gives the conversion factor for miles to kilometres to 7 significant figures.
    "We got the distance between the two stations by road from \href{http://theaa.com}{theaa.com}. We then converted this distance to kilometres by multiplying it by \(1.609344 . "\) Suppose this conversion factor is applied to a distance of exactly 100 miles.
    State which one of the following best expresses the level of accuracy for the distance in metric units, justifying your answer. A : to the nearest millimetre
    B : to the nearest 10 centimetres
    C : to the nearest metre