OCR MEI C2 (Core Mathematics 2) 2007 January

1 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).

2 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.

3 Given that \(\cos \theta = \frac { 1 } { 3 }\) and \(\theta\) is acute, find the exact value of \(\tan \theta\).

4 Sequences \(\mathrm { A } , \mathrm { B }\) and C are shown below. They each continue in the pattern established by the given terms. $$\begin{array} { l l l l l l l l l } \text { A: } & 1 , & 2 , & 4 , & 16 , & 32 , & \ldots & \\ \text { B: } & 20 , & - 10 , & 5 , & - 2.5 , & 1.25 , & - 0.625 , & \ldots \\ \text { C: } & 20 , & 5 , & 1 , & 20 , & 5 , & 1 , & \ldots \end{array}$$

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

5 A is the point \(( 2,1 )\) on the curve \(y = \frac { 4 } { x ^ { 2 } }\).
B is the point on the same curve with \(x\)-coordinate 2.1.

1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places.
2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A .
3. Use calculus to find the gradient of the curve at A .

6 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\).

8 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

9 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 8 x\). The curve passes through the point \(( 1,5 )\). Find the equation of the curve.

10

1. Express \(\log _ { a } x ^ { 4 } + \log _ { a } \left( \frac { 1 } { x } \right)\) as a multiple of \(\log _ { a } x\).
2. Given that \(\log _ { 10 } b + \log _ { 10 } c = 3\), find \(b\) in terms of \(c\).

11 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(A B , B C\) and \(C A\). The road AC runs due North and the measurements shown are in metres. \begin{figure}[h] \end{figure}

1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of the village green. The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_440_1002_1436_737} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} The new road is an arc of a circle with centre O and radius 130 m .
3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
   (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures.

12 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h] \end{figure}

1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \(^ { 2 }\).
3. Find the area of the region bounded by the curve and the \(x\)-axis.

13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\). The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.

1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
5. State one way in which this model is unrealistic.