OCR H240/02 (Pure Mathematics and Statistics) 2019 June

1

1. Differentiate the following.
   1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
   2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).

2

1. Find the coefficient of \(x ^ { 5 }\) in the expansion of \(( 3 - 2 x ) ^ { 8 }\).
2. 1. Expand \(( 1 + 3 x ) ^ { 0.5 }\) as far as the term in \(x ^ { 3 }\).
   2. State the range of values of \(x\) for which your expansion is valid. A student suggests the following check to determine whether the expansion obtained in part (b)(i) may be correct.
      "Use the expansion to find an estimate for \(\sqrt { 103 }\), correct to five decimal places, and compare this with the value of \(\sqrt { 103 }\) given by your calculator."
   3. Showing your working, carry out this check on your expansion from part (b)(i).

3

1. A circle is defined by the parametric equations \(x = 3 + 2 \cos \theta , y = - 4 + 2 \sin \theta\).
   1. Find a cartesian equation of the circle.
   2. Write down the centre and radius of the circle.
2. In this question you must show detailed reasoning. The curve \(S\) is defined by the parametric equations \(x = 4 \cos t , y = 2 \sin t\). The line \(L\) is a tangent to \(S\) at the point given by \(t = \frac { 1 } { 6 } \pi\). Find where the line \(L\) cuts the \(x\)-axis.

4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years.
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\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}

1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
   The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
   1. using model A,
   2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
3. State what is suggested about the long-term food supply by
   1. model A,
   2. model B.

5
\includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-06_371_339_255_251} For a cone with base radius \(r\), height \(h\) and slant height \(l\), the following formulae are given.
Curved surface area, \(S = \pi r l\)
Volume, \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\)
A container is to be designed in the shape of an inverted cone with no lid. The base radius is \(r \mathrm {~m}\) and the volume is \(V \mathrm {~m} ^ { 3 }\). The area of the material to be used for the cone is \(4 \pi \mathrm {~m} ^ { 2 }\).

1. Show that \(V = \frac { 1 } { 3 } \pi \sqrt { 16 r ^ { 2 } - r ^ { 6 } }\).
2. In this question you must show detailed reasoning. It is given that \(V\) has a maximum value for a certain value of \(r\).
   Find the maximum value of \(V\), giving your answer correct to 3 significant figures.

6 Shona makes the following claim.
" \(n\) is an even positive integer greater than \(2 \Rightarrow 2 ^ { n } - 1\) is not prime"
Prove that Shona's claim is true.

7 In this question you must show detailed reasoning.
Use the substitution \(u = 6 x ^ { 2 } + x\) to solve the equation \(36 x ^ { 4 } + 12 x ^ { 3 } + 7 x ^ { 2 } + x - 2 = 0\).

8 The stem-and-leaf diagram shows the heights, in centimetres, of 17 plants, measured correct to the nearest centimetre. Key: 5 | 6 means 56

1. Find the median and inter-quartile range of these heights.
2. Calculate the mean and standard deviation of these heights.
3. State one advantage of using the median rather than the mean as a measure of average for these heights.

9

1. The masses, in grams, of plums of a certain kind have the distribution \(\mathrm { N } ( 55,18 )\).
   1. Find the probability that a plum chosen at random has a mass between 50.0 and 60.0 grams.
   2. The heaviest \(5 \%\) of plums are classified as extra large. Find the minimum mass of extra large plums.
   3. The plums are packed in bags, each containing 10 randomly selected plums. Find the probability that a bag chosen at random has a total mass of less than 530 g .
2. The masses, in grams, of apples of a certain kind have the distribution \(\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)\). It is given that half of the apples have masses between 62 g and 72 g . Determine \(\sigma\).

10 The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 . This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the 5\% significance level, whether the mean level of pollutant has changed.

11 A trainer was asked to give a lecture on population profiles in different Local Authorities (LAs) in the UK. Using data from the 2011 census, he created the following scatter diagram for 17 selected LAs. \begin{figure}[h] \end{figure} He selected the 17 LAs using the following method. The proportions of people aged 18 to 24 and aged 65+ in any Local Authority are denoted by \(P _ { \text {young } }\) and \(P _ { \text {senior } }\) respectively. The trainer used a spreadsheet to calculate the value of \(k = \frac { P _ { \text {young } } } { P _ { \text {senior } } }\) for each of the 348 LAs in the UK. He then used specific ranges of values of \(k\) to select the 17 LAs.

1. Estimate the ranges of values of \(k\) that he used to select these 17 LAs.
2. Using the 17 LAs the trainer carried out a hypothesis test with the following hypotheses.
   \(\mathrm { H } _ { 0 }\) : There is no linear correlation in the population between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\).
   \(\mathrm { H } _ { 1 }\) : There is negative linear correlation in the population between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\).
   He found that the value of Pearson's product-moment correlation coefficient for the 17 LAs is - 0.797 , correct to 3 significant figures.
   1. Use the table on page 9 to show that this value is significant at the \(1 \%\) level. The trainer concluded that there is evidence of negative linear correlation between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\) in the population.
   2. Use the diagram to comment on the reliability of this conclusion.
3. Describe one outstanding feature of the population in the areas represented by the points in the bottom right hand corner of the diagram.
4. The trainer's audience included representatives from several universities. Suggest a reason why the diagram might be of particular interest to these people. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Critical values of Pearson's product-moment correlation coefficient}

   \multirow{2}{*}{1-tail test 2-tail test}	5\%	2.5\%	1\%	0.5\%
   	10\%	5\%	2\%	1\%
   \(n\)
   1	-	-	-	-
   2	-	-	-	-
   3	0.9877	0.9969	0.9995	0.9999
   4	0.9000	0.9500	0.9800	0.9900
   5	0.8054	0.8783	0.9343	0.9587
   6	0.7293	0.8114	0.8822	0.9172
   7	0.6694	0.7545	0.8329	0.8745
   8	0.6215	0.7067	0.7887	0.8343
   9	0.5822	0.6664	0.7498	0.7977
   10	0.5494	0.6319	0.7155	0.7646
   11	0.5214	0.6021	0.6851	0.7348
   12	0.4973	0.5760	0.6581	0.7079
   13	0.4762	0.5529	0.6339	0.6835
   14	0.4575	0.5324	0.6120	0.6614
   15	0.4409	0.5140	0.5923	0.6411
   16	0.4259	0.4973	0.5742	0.6226
   17	0.4124	0.4821	0.5577	0.6055
   18	0.4000	0.4683	0.5425	0.5897
   19	0.3887	0.4555	0.5285	0.5751
   20	0.3783	0.4438	0.5155	0.5614
   21	0.3687	0.4329	0.5034	0.5487
   22	0.3598	0.4227	0.4921	0.5368
   23	0.3515	0.4132	0.4815	0.5256
   24	0.3438	0.4044	0.4716	0.5151
   25	0.3365	0.3961	0.4622	0.5052
   26	0.3297	0.3882	0.4534	0.4958
   27	0.3233	0.3809	0.4451	0.4869
   28	0.3172	0.3739	0.4372	0.4785
   29	0.3115	0.3673	0.4297	0.4705
   30	0.3061	0.3610	0.4226	0.4629

   \end{table} Turn over for questions 12 and 13

12 A random variable \(X\) has probability distribution defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 1,2,3,4,5 ,
0 & \text { otherwise, } \end{cases}$$ where \(k\) is a constant.

1. Show that \(\mathrm { P } ( X = 3 ) = 0.2\).
2. Show in a table the values of \(X\) and their probabilities.
3. Two independent values of \(X\) are chosen, and their total \(T\) is found.
   1. Find \(\mathrm { P } ( T = 7 )\).
   2. Given that \(T = 7\), determine the probability that one of the values of \(X\) is 2 .

13 It is known that \(26 \%\) of adults in the UK use a certain app. A researcher selects a random sample of 5000 adults in the UK. The random variable \(X\) is defined as the number of adults in the sample who use the app. Given that \(\mathrm { P } ( X < n ) < 0.025\), calculate the largest possible value of \(n\).