OCR MEI C2 (Core Mathematics 2) 2011 January

1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).

2 Find \(\int \left( 3 x ^ { 5 } + 2 x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\).

3 At a place where a river is 7.5 m wide, its depth is measured every 1.5 m across the river. The table shows the results. Use the trapezium rule with 5 strips to estimate the area of cross-section of the river.

4 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of

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

5 The second term of a geometric sequence is 6 and the fifth term is - 48 .
Find the tenth term of the sequence.
Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence.

6 The third term of an arithmetic progression is 24 . The tenth term is 3 .
Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive.

7 Simplify

1. \(\log _ { 10 } x ^ { 5 } + 3 \log _ { 10 } x ^ { 4 }\),
2. \(\log _ { a } 1 - \log _ { a } a ^ { b }\).

8 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$

9 Charles has a slice of cake; its cross-section is a sector of a circle, as shown in Fig. 9. The radius is \(r \mathrm {~cm}\) and the sector angle is \(\frac { \pi } { 6 }\) radians. He wants to give half of the slice to Jan. He makes a cut across the sector as shown. \begin{figure}[h] \end{figure} Show that when they each have half the slice, \(a = r \sqrt { \frac { \pi } { 6 } }\). Section B (36 marks)

10 \begin{figure}[h] \end{figure} A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10.

1. The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = - 2 x - \frac { 1 } { 4 }\).
3. The two tangents intersect at the point D . Find the \(y\)-coordinate of D . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-4_773_1027_255_557} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).

1. Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
2. Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.

12 The table shows the size of a population of house sparrows from 1980 to 2005. The 'red alert' category for birds is used when a population has decreased by at least \(50 \%\) in the previous 25 years.

1. Show that the information for this population is consistent with the house sparrow being on red alert in 2005. The size of the population may be modelled by a function of the form \(P = a \times 10 ^ { - k t }\), where \(P\) is the population, \(t\) is the number of years after 1980, and \(a\) and \(k\) are constants.
2. Write the equation \(P = a \times 10 ^ { - k t }\) in logarithmic form using base 10, giving your answer as simply as possible.
3. Complete the table and draw the graph of \(\log _ { 10 } P\) against \(t\), drawing a line of best fit by eye.
4. Use your graph to find the values of \(a\) and \(k\) and hence the equation for \(P\) in terms of \(t\).
5. Find the size of the population in 2015 as predicted by this model. Would the house sparrow still be on red alert? Give a reason for your answer.