OCR H240/02 (Pure Mathematics and Statistics) 2020 November

1

1. Differentiate the following with respect to \(x\).
   1. \(( 2 x + 3 ) ^ { 7 }\)
   2. \(x ^ { 3 } \ln x\)
2. Find \(\int \cos 5 x \mathrm {~d} x\).
3. Find the equation of the curve through \(( 1,3 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x - 5\).

2 Simplify fully \(\frac { 2 x ^ { 3 } + x ^ { 2 } - 7 x - 6 } { x ^ { 2 } - x - 2 }\).

3 In this question you should assume that \(- 1 < x < 1\).

1. For the binomial expansion of \(( 1 - x ) ^ { - 2 }\)
   1. find and simplify the first four terms,
   2. write down the term in \(x ^ { n }\).
2. Write down the sum to infinity of the series \(1 + x + x ^ { 2 } + x ^ { 3 } + \ldots\).
3. Hence or otherwise find and simplify an expression for \(2 + 3 x + 4 x ^ { 2 } + 5 x ^ { 3 } + \ldots\) in the form \(\frac { a - x } { ( b - x ) ^ { 2 } }\) where \(a\) and \(b\) are constants to be determined.

4 In this question you must show detailed reasoning.
Solve the equation \(3 \sin ^ { 4 } \phi + \sin ^ { 2 } \phi = 4\), for \(0 \leqslant \phi < 2 \pi\), where \(\phi\) is measured in radians.

5

1. Determine the set of values of \(n\) for which \(\frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) are positive integers. A 'Pythagorean triple' is a set of three positive integers \(a , b\) and \(c\) such that \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
2. Prove that, for the set of values of \(n\) found in part (a), the numbers \(n , \frac { n ^ { 2 } - 1 } { 2 }\) and \(\frac { n ^ { 2 } + 1 } { 2 }\) form a Pythagorean triple.

6 Prove that \(\sqrt { 2 } \cos \left( 2 \theta + 45 ^ { \circ } \right) \equiv \cos ^ { 2 } \theta - 2 \sin \theta \cos \theta - \sin ^ { 2 } \theta\), where \(\theta\) is measured in degrees.
\(7 \quad A\) and \(B\) are fixed points in the \(x - y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of

1. the quantity \(| \mathbf { a } - \mathbf { b } |\),
2. the vector \(\frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } )\). The circle \(P\) is the set of points with position vector \(\mathbf { p }\) in the \(x - y\) plane which satisfy \(\left| \mathbf { p } - \frac { 1 } { 2 } ( \mathbf { a } + \mathbf { b } ) \right| = \frac { 1 } { 2 } | \mathbf { a } - \mathbf { b } |\).
3. State, in terms of \(\mathbf { a }\) and \(\mathbf { b }\),
   1. the position vector of the centre of \(P\),
   2. the radius of \(P\). It is now given that \(\mathbf { a } = \binom { 2 } { - 1 } , \mathbf { b } = \binom { 4 } { 5 }\) and \(\mathbf { p } = \binom { x } { y }\).
4. Find a cartesian equation of \(P\).

8 The rate of change of a certain population \(P\) at time \(t\) is modelled by the equation \(\frac { \mathrm { d } P } { \mathrm {~d} t } = ( 100 - P )\). Initially \(P = 2000\).

1. Determine an expression for \(P\) in terms of \(t\).
2. Describe how the population changes over time.

9 The histogram shows information about the numbers of cars in five different price ranges, sold in one year at a car showroom.
\includegraphics[max width=\textwidth, alt={}, center]{4ba60d6b-c987-4f6b-8c55-8b594c90c854-06_922_1413_495_244} It is given that 66 cars in the price range \(\pounds 10000\) to \(\pounds 20000\) were sold.

1. Find the number of cars sold in the price range \(\pounds 50000\) to \(\pounds 90000\).
2. State the units of the frequency density.
3. Suggest one change that the management could make to the diagram so that it would provide more information.
4. Estimate the number of cars sold in the price range \(\pounds 50000\) to \(\pounds 60000\).

10 Pierre is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.

11 As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

1. Calculate estimates of the mean and standard deviation of these masses. The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim \mathrm {~N} ( 200,3600 )\).
2. Use the model to find \(\mathrm { P } ( 150 < X < 210 )\).
3. Use the model to determine \(x _ { 1 }\) such that \(\mathrm { P } \left( 160 < X < x _ { 1 } \right) = 0.6\), giving your answer correct to five significant figures. It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.
4. Use these results to show that the model may not be appropriate.
5. Suggest a different value of a parameter of the model in the light of these results.

12 In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5\% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

1. State appropriate null and alternative hypotheses for the test.
2. Determine the rejection region for the test.

13 Andy and Bev are playing a game.

• The game consists of three points.
• On each point, \(\mathrm { P } (\) Andy wins \() = 0.4\) and \(\mathrm { P } (\) Bev wins \() = 0.6\).
• If one player wins two consecutive points, then they win the game, otherwise neither player wins.
  1. Determine the probability of the following events.
     1. Andy wins the game.
     2. Neither player wins the game.

Andy and Bev now decide to play a match which consists of a series of games.

• In each game, if a player wins the game then they win the match.
• If neither player wins the game then the players play another game.
• Determine the probability that Andy wins the match.

14 Table 1 shows the numbers of usual residents in the age range 0 to 4 in 15 Local Authorities (LAs) in 2001 and 2011. The table also shows the increase in the numbers in this age group, and the same increase as a percentage. \begin{table}[h] \end{table} Fig. 2 shows the increase in each LA in raw numbers, and Fig. 3 shows the percentage increase in each LA. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. The Education Committees in these LAs need to plan for the provision of schools for pupils in their districts.
   1. Explain why, in this context, the increase is more important than the actual numbers.
   2. In which of the following LAs was there likely to have been the greatest need for extra teachers in the years following 2011: Bolton, Sefton, Tameside or Wigan? Give a reason for your answer.
   3. State an assumption about the populations needed to make your answer in part (ii) valid.
2. In two of the 15 LAs the proportion of young families is greater than in the other 13 LAs. Suggest, using only data from Fig. 2 and Fig. 3 and/or Table 1, which two LAs these are most likely to be.

15 In this question you must show detailed reasoning. The random variable \(X\) has probability distribution defined as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { 15 } { 64 } \times \frac { 2 ^ { x } } { x ! } & x = 2,3,4,5 ,
0 & \text { otherwise. } \end{cases}\)

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 15 } { 32 }\). The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
2. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 9\), determine the probability that at least one of these three values is equal to 2 . Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.
3. Determine the probability that she chooses exactly 10 values of \(X\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA