Gradient condition leads to trig equation

7 A curve is defined by the parametric equations $$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$ where \(0 \leqslant t \leqslant 2 \pi\). At each of the points \(P\) and \(Q\) on the curve, the gradient of the curve is \(\frac { 5 } { 2 }\).

1. Show that the values of \(t\) at \(P\) and \(Q\) satisfy the equation \(10 \cos t - 8 \sin t = 5\).
2. Express \(10 \cos t - 8 \sin t\) in the form \(R \cos ( t + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
3. Hence find the values of \(t\) at the points \(P\) and \(Q\).

7 \includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-10_657_792_269_664} The diagram shows the curve with parametric equations $$x = 3 \cos 2 \theta , \quad y = 4 \sin \theta ,$$ for \(\pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). Points \(P\) and \(Q\) lie on the curve. The gradient of the curve at \(P\) is 2 . The straight line \(3 x + y = 0\) meets the curve at \(Q\).

1. Find the value of \(\theta\) at \(P\), giving your answer correct to 3 significant figures.
2. Find the gradient of the curve at \(Q\), giving your answer correct to 3 significant figures.
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7 The parametric equations of a curve are $$x = 2 t - \sin 2 t , \quad y = 5 t + \cos 2 t$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). At the point \(P\) on the curve, the gradient of the curve is 2 .

1. Show that the value of the parameter at \(P\) satisfies the equation \(2 \sin 2 t - 4 \cos 2 t = 1\).
2. By first expressing \(2 \sin 2 t - 4 \cos 2 t\) in the form \(R \sin ( 2 t - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the coordinates of \(P\). Give each coordinate correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.
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   OCR is part of the \section*{ADVANCED GCE
   MATHEMATICS (MEI)} 4754B
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   Wednesday 9 June 2010 Afternoon \includegraphics[width=\textwidth]{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[width=\textwidth]{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 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

5 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 \(0 \leqslant \theta \leqslant 2 \pi\).

4 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 \(0 \leqslant \theta \leqslant 2 \pi\).

12 The parametric equations of a curve are given by \(x = 2 \cos \theta\) and \(y = 3 \sin \theta\) for \(0 \leq \theta < 2 \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). The tangents to the curve at the points P and Q pass through the point \(( 2,6 )\).
2. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2 \sin \theta + \cos \theta = 1\).
3. Find the values of \(\theta\) at the points \(P\) and \(Q\).

The parametric equations of a curve are given by \(x = 2\cos\theta\) and \(y = 3\sin\theta\) for \(0 \leq \theta < 2\pi\).

1. Find \(\frac{dy}{dx}\) in terms of \(\theta\). [2]

The tangents to the curve at the points P and Q pass through the point (2, 6).

1. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2\sin\theta + \cos\theta = 1\). [4]
2. Find the values of \(\theta\) at the points P and Q. [5]