OCR MEI C2 (Core Mathematics 2) 2005 January

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).

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

3 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4 Fig. 4 For triangle ABC shown in Fig. 4, calculate

1. the length of BC ,
2. the area of triangle ABC .

5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.

6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4
a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

7 \begin{figure}[h] \end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).

1. Find \(\theta\).
2. Hence find the perimeter of the sector.

8

1. Solve the equation \(10 ^ { x } = 316\).
2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).

9

1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
   \end{figure} Using this model,
   (A) find the greatest height of the tunnel,
   (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
   Hence estimate the volume of earth removed when the tunnel is re-shaped.

10 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).

1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).

11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made. The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.

1. Give a physical interpretation for the constant \(z _ { 0 }\).
2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
   Hence estimate the temperature of the drink 70 minutes after it is made.