OCR MEI C2 (Core Mathematics 2) 2006 June

Write down the values of \(\log_a a\) and \(\log_a (a^3)\). [2]

The first term of a geometric series is 8. The sum to infinity of the series is 10. Find the common ratio. [3]

\(\theta\) is an acute angle and \(\sin \theta = \frac{1}{4}\). Find the exact value of \(\tan \theta\). [3]

Find \(\int_1^2 \left( x^4 - \frac{3}{x^2} + 1 \right) dx\), showing your working. [5]

The gradient of a curve is given by \(\frac{dy}{dx} = 3 - x^2\). The curve passes through the point \((6, 1)\). Find the equation of the curve. [4]

A sequence is given by the following. $$u_1 = 3$$ $$u_{n+1} = u_n + 5$$

1. Write down the first 4 terms of this sequence. [1]
2. Find the sum of the 51st to the 100th terms, inclusive, of the sequence. [4]

1. Sketch the graph of \(y = \cos x\) for \(0° \leq x \leq 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leq x \leq 360°\). Label each graph clearly. [3]
2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leq x \leq 360°\). [2]

Given that \(y = 6x^3 + \sqrt{x} + 3\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [5]

Use logarithms to solve the equation \(5^{3x} = 100\). Give your answer correct to 3 decimal places. [4]

1. \includegraphics{figure_10_1} At a certain time, ship S is 5.2 km from lighthouse L on a bearing of 048°. At the same time, ship T is 6.3 km from L on a bearing of 105°, as shown in Fig. 10.1. For these positions, calculate
   1. the distance between ships S and T, [3]
   2. the bearing of S from T. [3]
2. \includegraphics{figure_10_2} Ship S then travels at 24 km h\(^{-1}\) anticlockwise along the arc of a circle, keeping 5.2 km from the lighthouse L, as shown in Fig. 10.2. Find, in radians, the angle \(\theta\) that the line LS has turned through in 26 minutes. Hence find, in degrees, the bearing of ship S from the lighthouse at this time. [5]

A cubic curve has equation \(y = x^3 - 3x^2 + 1\).

1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points. [5]
2. Show that the tangent to the curve at the point where \(x = -1\) has gradient 9. Find the coordinates of the other point, P, on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P. Show that the area of the triangle bounded by the normal at P and the \(x\)- and \(y\)-axes is 8 square units. [8]

Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10^{bt}\), where \(t\) is the time in years after 2000.

1. Show that, according to this model, the graph of \(\log_{10} P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. [3]
2. The table gives the data for the population from 2001 to 2005.

   Year	2001	2002	2003	2004	2005
   \(t\)	1	2	3	4	5
   \(P\)	7900	8800	10000	11300	12800

   Complete the table of values on the insert, and plot \(\log_{10} P\) against \(t\). Draw a line of best fit for the data. [3]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
4. Predict the population in 2008 according to this model. [2]