OCR MEI C2 (Core Mathematics 2) 2010 January

1 Find \(\int \left( x - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x\).

2 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 55th term of this sequence, showing your method.
2. Find the sum of the first 55 terms of the sequence.

3 You are given that \(\sin \theta = \frac { \sqrt { 2 } } { 3 }\) and that \(\theta\) is an acute angle. Find the exact value of \(\tan \theta\).

4 A sector of a circle has area \(8.45 \mathrm {~cm} ^ { 2 }\) and sector angle 0.4 radians. Calculate the radius of the sector.

5 \begin{figure}[h] \end{figure} Fig. 5 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { P } , \mathrm { Q }\) and R .

1. \(y = \mathrm { f } ( 2 x )\)
2. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)

6

1. Find the 51 st term of the sequence given by $$\begin{aligned} u _ { 1 } & = 5
   u _ { n + 1 } & = u _ { n } + 4 \end{aligned}$$
2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-3_531_969_744_589} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure} Fig. 7 shows triangle ABC , with \(\mathrm { AB } = 8.4 \mathrm {~cm}\). D is a point on AC such that angle \(\mathrm { ADB } = 79 ^ { \circ }\), \(\mathrm { BD } = 5.6 \mathrm {~cm}\) and \(\mathrm { CD } = 7.8 \mathrm {~cm}\). Calculate
3. angle BAD ,
4. the length BC .

8 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).

9

1. Sketch the graph of \(y = 3 ^ { x }\).
2. Use logarithms to solve \(3 ^ { 2 x + 1 } = 10\), giving your answer correct to 2 decimal places.

10

1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
3. Sketch the curve.

11 Fig. 11 shows the cross-section of a school hall, with measurements of the height in metres taken at 1.5 m intervals from O . \begin{figure}[h] \end{figure}

1. Use the trapezium rule with 8 strips to calculate an estimate of the area of the cross-section.
2. Use 8 rectangles to calculate a lower bound for the area of the cross-section. The curve of the roof may be modelled by \(y = - 0.013 x ^ { 3 } + 0.16 x ^ { 2 } - 0.082 x + 2.4\), where \(x\) metres is the horizontal distance from O across the hall, and \(y\) metres is the height.
3. Use integration to find the area of the cross-section according to this model.
4. Comment on the accuracy of this model for the height of the hall when \(x = 7.5\).

12 Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005. Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a 10 ^ { b t }\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.

1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis.
2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log _ { 10 } P\) against \(t\). Draw, by eye, a line of best fit on your graph.
3. Use your graph to find the equation for \(P\) in terms of \(t\).
4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate.