OCR MEI C4 (Core Mathematics 4) 2009 June

1 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence solve the equation \(4 \cos \theta - \sin \theta = 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

2 Using partial fractions, find \(\int \frac { x } { ( x + 1 ) ( 2 x + 1 ) } \mathrm { d } x\).
[0pt] [7]

4 The part of the curve \(y = 4 - x ^ { 2 }\) that is above the \(x\)-axis is rotated about the \(y\)-axis. This is shown in Fig. 4. Find the volume of revolution produced, giving your answer in terms of \(\pi\). \begin{figure}[h] \end{figure}

5 A curve has parametric equations $$x = a t ^ { 3 } , \quad y = \frac { a } { 1 + t ^ { 2 } }$$ where \(a\) is a constant.
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 t \left( 1 + t ^ { 2 } \right) ^ { 2 } }\).
Hence find the gradient of the curve at the point \(\left( a , \frac { 1 } { 2 } a \right)\).

6 Given that \(\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3\), show that \(\cot ^ { 2 } \theta - \cot \theta - 2 = 0\).
Hence solve the equation \(\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\). Section B (36 marks)

7 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point \(\mathrm { A } ( 1,2,2 )\), and enters a glass object at point \(\mathrm { B } ( 0,0,2 )\). The surface of the glass object is a plane with normal vector \(\mathbf { n }\). Fig. 7 shows a cross-section of the glass object in the plane of the light ray and \(\mathbf { n }\). \begin{figure}[h] \end{figure}

1. Find the vector \(\overrightarrow { \mathrm { AB } }\) and a vector equation of the line AB . The surface of the glass object is a plane with equation \(x + z = 2\). AB makes an acute angle \(\theta\) with the normal to this plane.
2. Write down the normal vector \(\mathbf { n }\), and hence calculate \(\theta\), giving your answer in degrees. The line BC has vector equation \(\mathbf { r } = \left( \begin{array} { l } 0
   0
   2 \end{array} \right) + \mu \left( \begin{array} { l } - 2
   - 2
   - 1 \end{array} \right)\). This line makes an acute angle \(\phi\) with the normal to the plane.
3. Show that \(\phi = 45 ^ { \circ }\).
4. Snell's Law states that \(\sin \theta = k \sin \phi\), where \(k\) is a constant called the refractive index. Find \(k\). The light ray leaves the glass object through a plane with equation \(x + z = - 1\). Units are centimetres.
5. Find the point of intersection of the line BC with the plane \(x + z = - 1\). Hence find the distance the light ray travels through the glass object. \section*{[Question 8 is printed overleaf.]} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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8 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).

1. Fig. 8.1 shows a regular 12 -sided polygon inscribed in a circle of radius 1 unit, centre \(\mathrm { O } . \mathrm { AB }\) is one of the sides of the polygon. \(C\) is the midpoint of \(A B\). Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-4_455_428_523_900} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
   \end{figure} (A) Show that \(\mathrm { AB } = 2 \sin 15 ^ { \circ }\).
   (B) Use a double angle formula to express \(\cos 30 ^ { \circ }\) in terms of \(\sin 15 ^ { \circ }\). Using the exact value of \(\cos 30 ^ { \circ }\), show that \(\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }\).
   (C) Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6 \sqrt { 2 - \sqrt { 3 } }\).
2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-4_456_428_1621_900} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} (A) Show that \(\mathrm { DE } = 2 \tan 15 ^ { \circ }\).
   (B) Let \(t = \tan 15 ^ { \circ }\). Use a double angle formula to express \(\tan 30 ^ { \circ }\) in terms of \(t\). Hence show that \(t ^ { 2 } + 2 \sqrt { 3 } t - 1 = 0\).
   (C) Solve this equation, and hence show that \(\pi < 12 ( 2 - \sqrt { 3 } )\).
3. Use the results in parts (i)( \(C\) ) and (ii)( \(C\) ) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form. \section*{ADVANCED GCE
   MATHEMATICS (MEI)} 4754B
   Applications of Advanced Mathematics (C4) Paper B: Comprehension Candidates answer on the question paper
   Monday 1 June 2009
   OCR Supplied Materials:
   Morning
   • Insert (inserted)
   • MEI Examination Formulae and Tables (MF2)
   Duration: Up to 1 hour
   Other Materials Required:
   • Rough paper
     \includegraphics[max width=\textwidth, alt={}, center]{b4861178-720d-4803-a608-abef350efb0e-5_122_442_1023_1370}
   1 On lines 90 and 91, the article says "The average score for each player works out to be 0.25 points per round". Derive this figure. 2 Line 47 gives the inequality \(b > c > d > w\).
   Interpret each of the following inequalities in the context of the example from the 1st World War.
4. \(b > w\)
5. \(c > d\)
6. \(\_\_\_\_\)
7. \(\_\_\_\_\)
   3 Table 3 illustrates a possible game where you always co-operate. In lines 98 and 99 the article says "Clearly the longer the game goes on the closer your average score approaches - 2 points per round and that of your opponent approaches 3 ." How many rounds have you played when your average score is - 1.999 ?
   4 A Prisoner's Dilemma game is proposed in which $$b = 6 , c = 1 , d = - 1 \text { and } w = - 3 .$$ Using the information in the article, state whether these values would allow long-term co-operation to evolve. Justify your answer.
   5 In a Prisoner's Dilemma game both players keep strictly to a Tit-for-tat strategy. You start with C and your opponent starts with D . The scoring system of \(b = 3 , c = 1 , d = - 1\) and \(w = - 2\) is used.
8. This table shows the first 8 out of many rounds. Complete the table.

   Round	You	Opponent	Your score	Opponent's score
   1	C	D
   2
   3
   4
   5
   6
   7
   8
   ⋯	⋯	⋯	⋯	⋯

9. Find your average score per round in the long run.
   6 In the article, the scoring system is \(b = 3 , c = 1 , d = - 1\) and \(w = - 2\). In Axelrod's experiment, negative numbers were avoided by taking \(b = 5 , c = 3 , d = 1\) and \(w = 0\). State the effect this change would have on
10. the players' scores,
11. who wins.
12. \(\_\_\_\_\)
13. \(\_\_\_\_\)
    7 Two companies, X and Y , are the only sellers of ice cream on an island. They both have a market share of about \(50 \%\). Although their ice cream is much the same, both companies spend a lot of money on advertising.
14. What agreement might the companies reach if they decide to co-operate?
15. What advantage would a company hope to gain by 'defecting' from this agreement?
    RECOGNISING ACHIEVEMENT