OCR MEI C4 2010 June — Question 5

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Year2010
SessionJune
TopicParametric equations
  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.
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    MATHEMATICS (MEI)} 4754B
    Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
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    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. \(\_\_\_\_\)
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