OCR MEI C2 (Core Mathematics 2) 2008 June

1 Express \(\frac { 7 \pi } { 6 }\) radians in degrees.

3 State the transformation which maps the graph of \(y = x ^ { 2 } + 5\) onto the graph of \(y = 3 x ^ { 2 } + 15\).

4 Use calculus to find the set of values of \(x\) for which \(\mathrm { f } ( x ) = 12 x - x ^ { 3 }\) is an increasing function.

5 In Fig. 5, A and B are the points on the curve \(y = 2 ^ { x }\) with \(x\)-coordinates 3 and 3.1 respectively. \begin{figure}[h] \end{figure}

1. Find the gradient of the chord AB . Give your answer correct to 2 decimal places.
2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2 ^ { x }\) at A .

6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x }\). Find the equation of the curve, given that it passes through the point \(( 9,105 )\). \begin{figure}[h] \end{figure} A sector of a circle of radius 6 cm has angle 1.6 radians, as shown in Fig. 7.
Find the area of the sector.
Hence find the area of the shaded segment.

8 The 11th term of an arithmetic progression is 1 . The sum of the first 10 terms is 120 . Find the 4th term. [5]

9 Use logarithms to solve the equation \(5 ^ { x } = 235\), giving your answer correct to 2 decimal places.

10 Showing your method, solve the equation \(2 \sin ^ { 2 } \theta = \cos \theta + 2\) for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

11 \begin{figure}[h] \end{figure} Fig. 11 shows a sketch of the cubic curve \(y = \mathrm { f } ( x )\). The values of \(x\) where it crosses the \(x\)-axis are - 5 , - 2 and 2 , and it crosses the \(y\)-axis at \(( 0 , - 20 )\).

1. Express \(\mathrm { f } ( x )\) in factorised form.
2. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-5_689_1006_269_568} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
5. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough.
6. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15\), for \(0 \leqslant x \leqslant 0.5\). Calculate \(\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x\) and state what this represents.
   Hence find the volume of water in the trough as given by this model.

13 The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years. Source: Department of National Statistics, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
This growth may be modelled by an equation of the form $$P = a t ^ { b } ,$$ where \(P\) is the percentage of the adult population visiting the cinema, \(t\) is the number of years after the year 1985/86 and \(a\) and \(b\) are constants to be determined.

1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(\log _ { 10 } t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. \section*{Answer part (ii) of this question on the insert provided.}
2. Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(\log _ { 10 } t\). Draw by eye a line of best fit for the data.
3. Use your graph to find the equation for \(P\) in terms of \(t\).
4. Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when \(t = 22\) ), according to this model.