OCR MEI C2 (Core Mathematics 2) 2008 June

Express \(\frac{7\pi}{6}\) radians in degrees. [2]

The first term of a geometric series is 5.4 and the common ratio is 0.1.

1. Find the fourth term of the series. [1]
2. Find the sum to infinity of the series. [2]

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

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

In Fig. 5, A and B are the points on the curve \(y = 2^x\) with \(x\)-coordinates 3 and 3.1 respectively. \includegraphics{figure_5}

1. Find the gradient of the chord AB. Give your answer correct to 2 decimal places. [2]
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. [2]

A curve has gradient given by \(\frac{\text{d}y}{\text{d}x} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]

\includegraphics{figure_7} 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. [5]

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

Use logarithms to solve the equation \(5^x = 235\), giving your answer correct to 2 decimal places. [3]

Showing your method, solve the equation \(2\sin^2\theta = \cos\theta + 2\) for values of \(\theta\) between \(0°\) and \(360°\). [5]

\includegraphics{figure_11} Fig. 11 shows a sketch of the cubic curve \(y = \text{f}(x)\). 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 f(\(x\)) in factorised form. [2]
2. Show that the equation of the curve may be written as \(y = x^3 + 5x^2 - 4x - 20\). [2]
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. [6]
4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \text{f}(2x)\). [2]

\includegraphics{figure_12} 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.

1. 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. [5]
2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8x^3 - 3x^2 - 0.5x - 0.15\), for \(0 \leq x \leq 0.5\). Calculate \(\int_0^{0.5} (8x^3 - 3x^2 - 0.5x - 0.15) \, \text{d}x\) and state what this represents. Hence find the volume of water in the trough as given by this model. [7]

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, www.statistics.gov.uk This growth may be modelled by an equation of the form $$P = at^b,$$ 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. [3]
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. [4]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
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. [1]