OCR MEI C2 (Core Mathematics 2) 2008 January

1 Differentiate \(10 x ^ { 4 } + 12\).

2 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

3 You are given that \(\tan \theta = \frac { 1 } { 2 }\) and the angle \(\theta\) is acute. Show, without using a calculator, that \(\cos ^ { 2 } \theta = \frac { 4 } { 5 }\).

4 \begin{figure}[h] \end{figure} Fig. 4 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 \(\mathrm { A } , \mathrm { B }\) and C .

1. \(y = 2 \mathrm { f } ( x )\)
2. \(y = \mathrm { f } ( x + 3 )\)

5 Find \(\int \left( 12 x ^ { 5 } + \sqrt [ 3 ] { x } + 7 \right) \mathrm { d } x\).

6

1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\). Give your answers in the form \(k \pi\).

7

1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
3. Sketch the curve with equation \(y = 2 ^ { x }\).

8 The second term of a geometric progression is 18 and the fourth term is 2 . The common ratio is positive. Find the sum to infinity of this progression.

9 You are given that \(\log _ { 10 } y = 3 x + 2\).

1. Find the value of \(x\) when \(y = 500\), giving your answer correct to 2 decimal places.
2. Find the value of \(y\) when \(x = - 1\).
3. Express \(\log _ { 10 } \left( y ^ { 4 } \right)\) in terms of \(x\).
4. Find an expression for \(y\) in terms of \(x\). Section B (36 marks)

10 \begin{figure}[h] \end{figure} Fig. 10 shows a solid cuboid with square base of side \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\). Its volume is \(120 \mathrm {~cm} ^ { 3 }\).

1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the cuboid is given by \(A = 2 x ^ { 2 } + \frac { 480 } { x }\).
2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} x ^ { 2 } }\).
3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case.

11

1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_661_869_404_680} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} (A) Calculate the total length of the course for this race.
   (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC.
2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_709_821_1603_703} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} Calculate the total length of the course for this race.

12

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
   (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (D) State what this limit represents.