OCR MEI AS Paper 2 (AS Paper 2) 2019 June

1 Solve the equation \(4 x ^ { - \frac { 1 } { 2 } } = 7\), giving your answer as a fraction in its lowest terms.

2 Fig. 2 shows a triangle with one angle of \(117 ^ { \circ }\) given. The lengths are given in centimetres. \begin{figure}[h] \end{figure} Calculate the area of the triangle, giving your answer correct to 3 significant figures.

3 Without using a calculator, prove that \(3 \sqrt { 2 } > 2 \sqrt { 3 }\).

4 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 8 x - 6 y - 39 = 0\).

1. Find the coordinates of the centre of the circle.
2. Find the radius of the circle.

5 Each day John either cycles to work or goes on the bus.

• If it is raining when John is ready to set off for work, the probability that he cycles to work is 0.4.
• If it is not raining when John is ready to set off for work, the probability that he cycles to work is 0.9 .
• The probability that it is raining when he is ready to set off for work is 0.2 .

You should assume that days on which it rains occur randomly and independently.

1. Draw a tree diagram to show the possible outcomes and their associated probabilities.
2. Calculate the probability that, on a randomly chosen day, John cycles to work. John works 5 days each week.
3. Calculate the probability that he cycles to work every day in a randomly chosen working week.

6 The large data set gives information about life expectancy at birth for males and females in different London boroughs. Fig. 6.1 shows summary statistics for female life expectancy at birth for the years 2012-2014. Fig. 6.2 shows summary statistics for male life expectancy at birth for the years 2012-2014. \section*{Female Life Expectancy at Birth} \begin{table}[h] \end{table} Male Life Expectancy at Birth \begin{table}[h] \end{table}

1. Use the information in Fig. 6.1 and Fig. 6.2 to draw two box plots. Draw one box plot for female life expectancy at birth in London boroughs and one box plot for male life expectancy at birth in London boroughs.
2. Compare and contrast the distribution of male life expectancy at birth with the distribution of female life expectancy at birth in London boroughs in 2012-2014. Lorraine, who lives in Lancashire, says she wishes her daughter (who was born in 2013) had been born in the London borough of Barnet, because her daughter would have had a higher life expectancy.
3. Give two reasons why there is no evidence in the large data set to support Lorraine's comment.
4. Use the mean and standard deviation for the summary statistics given in Fig. 6.1 and Fig. 6.2 to show that there is at least one outlier in each set. The scatter diagram in Fig. 6.3 shows male life expectancy at birth plotted against female life expectancy at birth for London boroughs in 2012-14. The outliers have been removed. Male life expectancy at birth against female life expectancy at birth \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{11e5167f-9f95-4494-9b66-b59fdce8b1ef-5_593_1054_1260_246} \captionsetup{labelformat=empty} \caption{Fig. 6.3}
   \end{figure}
5. Describe the association between male life expectancy at birth and female life expectancy at birth in London boroughs in 2012-14.

7

1. Find \(\int x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x\).
2. Show that \(\int _ { 0 } ^ { 8 } x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x = a \times 2 ^ { 11 }\), where \(a\) is a positive integer to be determined.

8 According to the latest research there are 19.8 million male drivers and 16.2 million female drivers on the roads in the UK.

1. A driver in the UK is selected at random. Find the probability that the driver is male.
2. Calculate the probability that there are 7 female drivers in a random sample of 25 UK drivers. When driving in a built-up area, Rebecca exceeded the speed limit and was obliged to attend a speed awareness course. Her husband said "It's nearly always male drivers who are speeding." When Rebecca attends the course, she finds that there are 25 drivers, 7 of whom are female. You should assume that the drivers on the speed awareness course constitute a random sample of drivers caught speeding.
3. In this question you must show detailed reasoning. Conduct a hypothesis test to determine whether there is any evidence at the \(5 \%\) level to suggest that male drivers are more likely to exceed the speed limit than female drivers.
4. State a modelling assumption that is necessary in order to conduct the hypothesis test in part (c).

9 In 2012 Adam bought a second hand car for \(\pounds 8500\). Each year Adam has his car valued. He believes that there is a non-linear relationship between \(t\), the time in years since he bought the car, and \(V\), the value of the car in pounds. Fig. 9.1 shows successive values of \(V\) and \(\log _ { 10 } V\). \begin{table}[h] \end{table} Adam uses a spreadsheet to plot the points ( \(t , \log _ { 10 } V\) ) shown in Fig. 9.1, and then generates a line of best fit for these points. The line passes through the points \(( 0,3.93 )\) and \(( 4,3.58 )\). A copy of his graph is shown in Fig. 9.2. \begin{figure}[h] \end{figure}

1. Find an expression for \(\log _ { 10 } V\) in terms of \(t\).
2. Find a model for \(V\) in the form \(V = A \times b ^ { t }\), where \(A\) and \(b\) are constants to be determined. Give the values of \(A\) and \(b\) correct to 2 significant figures. In 2017 Adam's car was valued at \(\pounds 3150\).
3. Determine whether the model is a good fit for this data. A company called Webuyoldcars pays \(\pounds 500\) for any second hand car. Adam decides that he will sell his car to this company when the annual valuation of his car is less than \(\pounds 500\).
4. According to the model, after how many years will Adam sell his car to Webuyoldcars?

10 In this question you must show detailed reasoning. The equation of a curve is \(y = \frac { x ^ { 2 } } { 4 } + \frac { 2 } { x } + 1\). A tangent and a normal to the curve are drawn at the point where \(x = 2\). Calculate the area bounded by the tangent, the normal and the \(x\)-axis. \section*{END OF QUESTION PAPER}