OCR MEI AS Paper 2 (AS Paper 2) 2023 June

1 A researcher collects data concerning the number of different social media platforms used by school pupils on a typical weekday. The frequency table for the data is shown below. The researcher uses software to represent the results in this diagram.
\includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-04_961_1195_737_242}

1. Explain why this diagram is inappropriate.
2. Calculate the following for the number of social media platforms used:
   1. the mean,
   2. the standard deviation.

2

1. Express \(x ^ { 2 } - 6 x + 1\) in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } - \mathrm { b }\), where \(a\) and \(b\) are integers to be determined.
2. Hence state the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 1\).

3 A student makes the following conjecture.
For all positive integers \(n , 6 n - 1\) is always prime. Use a counter example to disprove this conjecture.

4 The equation of a curve is \(\mathrm { y } = \frac { \mathrm { k } } { \mathrm { x } ^ { 2 } }\), where \(k\) is a constant.
The curve passes through the point \(( 2,1 )\).

1. Find the value of \(k\).
2. Sketch the curve.

5 Show that the distance between the points \(( 5,2 )\) and \(( 11 , - 1 )\) is \(a \sqrt { b }\), where \(a\) and \(b\) are integers to be determined.

6 An app on my new smartphone records the number of times in a day I use the phone. The data for each day since I bought the phone are shown in the stem and leaf diagram. Key: 3|1 means 31

1. Explain whether these data are a sample or a population.
2. Describe the shape of the distribution.
3. Determine the interquartile range.
4. Use your answer to part (c) to determine whether there are any outliers in the lower tail.

7

1. Use the factor theorem to show that \(( x - 2 )\) is a factor of \(x ^ { 3 } + 6 x ^ { 2 } - x - 30\).
2. Factorise \(x ^ { 3 } + 6 x ^ { 2 } - x - 30\) completely.

8 The pre-release material contains information on Pulse Rate and Body Mass Index (BMI). A student is investigating whether there is a relationship between pulse rate and BMI. A section of the available data is shown in the table. The student decides to draw a scatter diagram.

1. With reference to the table, explain which data should be cleaned before any analysis takes place. The student cleans the data for BMI and Pulse Rate in the pre-release material and draws a scatter diagram. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Scatter diagram of Pulse Rate against BMI} \includegraphics[alt={},max width=\textwidth]{82438df0-6550-4ffd-92d8-3c67bec59a6b-06_869_1575_1585_246}
   \end{figure} The student identifies one outlier.
2. On the copy of the scatter diagram in the Printed Answer Booklet, circle this outlier. The student decides to remove this outlier from the data. They then use the LINEST function in the spreadsheet to obtain the following formula for the line of best fit.
   \(\mathrm { P } = 0.29 \mathrm { Q } + 64.2\),
   where \(P =\) PulseRate and \(Q = \mathrm { BMI }\). They use this to estimate the Pulse Rate of a person with BMI 23.68.
   They obtain a value of 71 correct to the nearest whole number.
3. With reference to the scatter diagram, explain whether it is appropriate to use the formula for the line of best fit. It is suggested that all pairs of values where the pulse rate is above 100 should also be cleaned from the data, as they must be incorrect.
4. Use your knowledge of the pre-release material to explain whether or not all pairs of values with a pulse rate of more than 100 should be cleaned from the data.

9 The table shows the probability distribution for the discrete random variable \(X\). You are given that \(q\) is a positive constant.

1. Determine the value of \(q\).
2. Calculate \(\mathrm { P } ( X \leqslant 4 )\). Two independent values of \(X\) are taken.
3. Determine the probability that the sum of the two values is 3 . Fifty independent values of \(X\) are taken.
4. Find the probability that a value of 2 occurs exactly 17 times.

10 In this question you must show detailed reasoning.
The diagram shows triangle ABC , where \(\mathrm { AB } = 3.9 \mathrm {~cm} , \mathrm { BC } = 4.5 \mathrm {~cm}\) and \(\mathrm { AC } = 3.5 \mathrm {~cm}\). Determine the area of triangle ABC .

11 In this question you must show detailed reasoning.
The equation of a curve is \(y = 2 x ^ { 3 } + 9 x ^ { 2 } + 24 x - 8\).
Show that there are no stationary points on this curve.

12 Doctors are investigating the weights of adult males registered at their surgery. One week they collect a sample by noting the weight in kilograms of all the adult males who have an appointment at their surgery.

1. State the sampling method they use.
2. Explain why this method will not generate a simple random sample of all the adult males registered at their surgery. They represent the data using a histogram.
   \includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-09_1166_1243_726_233} An incomplete frequency table for the data is shown below.

   Weight in kg	\(50 -\)	\(65 -\)	\(75 -\)	\(80 -\)	\(90 -\)	\(100 - 120\)
   Frequency		8

3. Complete the copy of the frequency table in the Printed Answer Booklet. One of these patients is selected at random.
4. Determine an estimate of the probability that he weighs either less than 60 kg or more than 110 kg .
5. Explain why your answer to part (d) is an estimate and not exact.

13 In a report published in October 2021 it is stated that \(37 \%\) of adults in the United Kingdom never exercise or play sport. A researcher believes that the true percentage is less than this. They decide to carry out a hypothesis test at the \(5 \%\) level to investigate the claim.

1. State the null and alternative hypotheses for their test.
2. Define the parameter for their test. In a random sample of 118 adults, they find that 35 of them never exercise or play sport.
3. Carry out the test.

14 In this question you must show detailed reasoning. The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
Determine the equation of the tangent to the curve at the point where \(x = 4\).

15 A family is planning a holiday in Europe. They need to buy some euros before they go. The exchange rate, \(y\), is the number of euros they can buy per pound. They believe that the exchange rate may be modelled by the formula
\(y = a t ^ { 2 } + b t + c\),
where \(t\) is the time in days from when they first check the exchange rate.
Initially, when \(t = 0\), the exchange rate is 1.14 .

1. Write down the value of \(c\). When \(t = 2 , y = 1.20\) and when \(t = 4 , y = 1.25\).
2. Calculate the values of \(a\) and \(b\). The family will only buy their euros when their model predicts an exchange rate of at least 1.29 .
3. Determine the range of values of \(t\) for which, according to their model, they will buy their euros.
4. Explain why the family's model is not viable in the long run.