OCR MEI AS Paper 2 (AS Paper 2) Specimen

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

2

1. Express \(2 \log _ { 3 } x + \log _ { 3 } a\) as a single logarithm.
2. Given that \(2 \log _ { 3 } x + \log _ { 3 } a = 2\), express \(x\) in terms of \(a\).

3 Show that the area of the region bounded by the curve \(y = 3 x ^ { - \frac { 3 } { 2 } }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis is \(6 - 2 \sqrt { 3 }\).

4 There are four human blood groups; these are called \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and AB . Each person has one of these blood groups. The table below shows the distribution of blood groups in a large country. Two people are selected at random from this country.

1. Find the probability that at least one of these two people has blood group O .
2. Find the probability that each of these two people has a different blood group.

5 A triangular field has sides of length \(100 \mathrm {~m} , 120 \mathrm {~m}\) and 135 m .

1. Find the area of the field.
2. Explain why it would not be reasonable to expect your answer in (a) to be accurate to the nearest square metre.

6

1. The graph of \(y = 3 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) is shown in Fig. 6.
   On the copy of Fig. 6 in the Printed Answer Booklet, sketch the graph of \(y = 2 \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-05_818_1507_571_351} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. In this question you must show detailed reasoning. Determine the values of \(\theta , 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\), for which the two graphs cross.

7 A farmer has 200 apple trees. She is investigating the masses of the crops of apples from individual trees. She decides to select a sample of these trees and find the mass of the crop for each tree.

1. Explain how she can select a random sample of 10 different trees from the 200 trees. The masses of the crops from the 10 trees, measured in kg, are recorded as follows.
   \(\begin{array} { l l l l l l l l l l } 23.5 & 27.4 & 26.2 & 29.0 & 25.1 & 27.4 & 26.2 & 28.3 & 38.1 & 24.9 \end{array}\)
2. For these data find
   • the mean,
   • the sample standard deviation.
   • Show that there is one outlier at the upper end of the data. How should the farmer decide whether to use this outlier in any further analysis of the data?

8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes. Fig. 8 shows the experimental data. \begin{figure}[h] \end{figure} It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.

1. Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
2. Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
3. Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
   1. \(\quad t = 0\),
   2. \(t = 20\).
4. Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.

9 Fig. 9.1 shows box and whisker diagrams which summarise the birth rates per 1000 people for all the countries in three of the regions as given in the pre-release data set.
The diagrams were drawn as part of an investigation comparing birth rates in different regions of the world. Africa (Sub-Saharan)
\includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_104_991_557_730} East and South East Asia
\includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_109_757_744_671} Caribbean
\includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_99_369_982_730} \begin{figure}[h] \end{figure}

1. Discuss the distributions of birth rates in these regions of the world. Make three different statements. You should refer to both information from the box and whisker diagrams and your knowledge of the large data set.
2. The birth rates for all the countries in Australasia are shown below.

   Country	Birth rate per 1000
   Australia	12.19
   New Zealand	13.4
   Papua New Guinea	24.89

   1. Explain why the calculation below is not a correct method for finding the birth rate per 1000 for Australasia as a whole. $$\frac { 12.19 + 13.4 + 24.89 } { 3 } \approx 16.83$$
   2. Without doing any calculations, explain whether the birth rate per 1000 for Australasia as a whole is higher or lower than 16.83 . The scatter diagram in Fig. 9.2 shows birth rate per 1000 and physicians/ 1000 population for all the countries in the pre-release data set. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-09_898_1698_386_274} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
      \end{figure}
3. Describe the correlation in the scatter diagram.
4. Discuss briefly whether the scatter diagram shows that high birth rates would be reduced by increasing the number of physicians in a country.

10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.

1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
   1. exactly 28 trains arrive on time,
   2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,
   • state the hypotheses,
   • find the critical region.

11 In this question you must show detailed reasoning. Fig. 11 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic function. Fig. 11 also shows the coordinates of the turning points and the points of intersection with the axes. \begin{figure}[h] \end{figure} Show that the tangent to \(y = \mathrm { f } ( x )\) at \(x = t\) is parallel to the tangent to \(y = \mathrm { f } ( x )\) at \(x = - t\) for all values of \(t\).

12 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: Let \(\arcsin x = \theta\) ] \section*{END OF QUESTION PAPER}