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OCR Further Pure Core 1 2018 September Q9
5 marks Standard +0.8
9 The diagram below shows the curve \(r = 4 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{c03cae53-eb00-496b-948f-ccff676bc03c-3_311_775_1713_644}
  1. On the diagram in your Printed Answer Booklet, shade the region \(R\) for which $$r \leqslant 4 \sin 3 \theta \text { and } 0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi .$$ In this question you must show detailed reasoning.
  2. Find the exact area of the region \(R\).
OCR Further Pure Core 1 2018 September Q10
6 marks Standard +0.3
10
  1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
  2. Find the range of \(x\) for which the expansion is valid.
  3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$
OCR Further Pure Core 1 2018 September Q11
8 marks Standard +0.8
11 A particular radioactive substance decays over time.
A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t .$$
  1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). Initially there was 10 g of the substance.
  2. Find the particular solution of the differential equation.
  3. Find to 6 significant figures the amount of substance that would be predicted by the model at
    1. 6 hours,
    2. 6.25 hours.
    3. Comment on the appropriateness of the model for predicting the amount of substance over time. \section*{END OF QUESTION PAPER}
OCR Further Statistics 2018 September Q1
4 marks Moderate -0.8
1 An experiment involves releasing a coin on a sloping plane so that it slides down the slope and then slides along a horizontal plane at the bottom of the slope before coming to rest. The angle \(\theta ^ { \circ }\) of the sloping plane is varied, and for each value of \(\theta\), the distance \(d \mathrm {~cm}\) the coin slides on the horizontal plane is recorded. A scatter diagram to illustrate the results of the experiment is shown below, together with the least squares regression line of \(d\) on \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{28c6a0d9-09a6-4743-af0e-fe2e43e256c9-2_639_972_561_548}
  1. State which two of the following correctly describe the variable \(\theta\).
    Controlled variableCorrelation coefficient
    Dependent variableIndependent variable
    Response variableRegression coefficient
    The least squares regression line of \(d\) on \(\theta\) has equation \(d = 1.96 + 0.11 \theta\).
  2. Use the diagram in the Printed Answer Booklet to explain the term "least squares".
  3. State what difference, if any, it would make to the equation of the regression line if \(d\) were measured in inches rather than centimetres. ( 1 inch \(\approx 2.54 \mathrm {~cm}\) ).
OCR Further Statistics 2018 September Q2
7 marks Standard +0.3
2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.
From tomorrow, I will count the number, \(X\), of nights on which I look for shooting stars, up to and including the first successful night. Find \(\mathrm { E } ( X )\).
OCR Further Statistics 2018 September Q3
7 marks Standard +0.8
3 A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\).
The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method, which should be justified, to find \(\mathrm { P } ( \bar { X } \leqslant 6.10 )\).
OCR Further Statistics 2018 September Q4
9 marks Standard +0.3
4 A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(x\) minutes, by an exponential distribution with probability density function (PDF) $$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ \section*{(i) In this question you must show detailed reasoning.} The mean waiting time is found to be 5.0 minutes. Show that \(\lambda = 0.2\).
(ii) Use the model to calculate the probability that a customer has to wait longer than 20 minutes for a response. In practice it is found that no customer waits for more than 15 minutes for a response. The statistician constructs an improved model to incorporate this fact.
(iii) On the diagram in the Printed Answer Booklet, sketch the following, labelling the curves clearly:
  1. the PDF of the model using the exponential distribution,
  2. a possible PDF for the improved model.
OCR Further Statistics 2018 September Q5
8 marks Standard +0.3
5 Hal designs a 4-edged spinner with edges labelled 1, 2, 3 and 4. He intends that the probability that the spinner will land on any edge should be proportional to the number on that edge. He spins the spinner 20 times and on each spin he records the number of the edge on which it lands. The results are shown in the table.
Edge number1234
Frequency3746
Test at the \(10 \%\) significance level whether the results are consistent with the intended probabilities.
OCR Further Statistics 2018 September Q6
10 marks Standard +0.8
6 A bag contains 7 red counters and 5 blue counters.
  1. Fred chooses 4 counters at random, without replacement. Show that the probability that Fred chooses exactly 2 red counters is \(\frac { 14 } { 33 }\).
  2. Lina chooses 4 counters at random from the bag, records whether or not exactly 2 red counters are chosen, and returns the counters to the bag. She carries out this experiment 99 times.
    1. Find the mean of the number of experiments that result in choosing exactly 2 red counters.
    2. Find the variance of the number of experiments that result in choosing exactly 2 red counters.
    3. Alex arranges all 12 counters in a random order in a straight line. A is the event: no two blue counters are next to one another. B is the event: all the blue counters are next to one another. Find \(\mathrm { P } ( A \cup B )\).
OCR Further Statistics 2018 September Q7
11 marks Standard +0.3
7 The table shows the values of 5 observations of bivariate data \(( x , y )\).
\(x\)4.65.96.57.88.3
\(y\)15.610.810.410.19.7
$$n = 5 , \Sigma x = 33.1 , \Sigma y = 56.6 , \Sigma x ^ { 2 } = 227.95 , \Sigma y ^ { 2 } = 664.26 , \Sigma x y = 362.37$$
  1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
  2. State what this value of \(r\) tells you about a scatter diagram illustrating the data.
  3. Test at the \(5 \%\) significance level whether there is association between \(x\) and \(y\).
  4. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
  5. State whether \(r , r _ { s }\), or both or neither is changed when the values of \(x\) are replaced by
    1. \(3 x - 2\),
    2. \(\sqrt { x }\).
OCR Further Statistics 2018 September Q8
8 marks Standard +0.3
8 In an experiment to investigate the effect of background music in carrying out work, ten students were each given a task. Five of the students did the task in silence and the other five did the task with background music. The scores on the tasks were as follows.
Silence4346555861
Background music1931385270
  1. Use a Wilcoxon rank-sum test to test at the 10\% level whether the presence of background music affects scores.
  2. A statistician suggests that the experiment is redesigned so that each student takes one task in silence and another task with background music. The differences in the test scores would then be analysed using a paired-sample method. State an advantage in redesigning the experiment in this way.
OCR Further Statistics 2018 September Q9
11 marks Standard +0.3
9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .
  1. Find an unbiased estimate of \(\mu\).
  2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\). \(D\) is the sum of 10 independent observations of \(C\).
  3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
  4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Discrete 2018 September Q1
7 marks Standard +0.3
1 The design for the lines on a playing area for a game is shown below. The letters are not part of the design. \includegraphics[max width=\textwidth, alt={}, center]{22571082-016b-409b-bfeb-e7ebf48ccac7-2_350_855_388_605} Priya paints the lines by pushing a machine. When she is pushing the machine she is about a metre behind the point being painted. She must not duplicate any line by painting it twice.
  • To relocate the machine, it must be stopped and then started again to continue painting the lines.
  • When the machine is being relocated it must still be pushed along the lines of the design, and not 'cut across' on a diagonal for example.
  • The machine can be turned through \(90 ^ { \circ }\) without having to be stopped.
    1. What is the minimum number of times that the machine will need to be started to paint the design?
The design is horizontally and vertically symmetric. $$\mathrm { AB } = 6 \text { metres, } \mathrm { AE } = 26 \text { metres, } \mathrm { AF } = 1.5 \text { metres and } \mathrm { AS } = 9 \text { metres. }$$
  • (a) Find the minimum distance that Priya needs to walk to paint the design. You should show enough working to make your reasoning clear but you do not need to use an algorithmic method.
    (b) Why, in practice, will the distance be greater than this?
    (c) What additional information would you need to calculate a more accurate shortest distance?
    •
    OCR Further Discrete 2018 September Q2
    8 marks Standard +0.8
    2 A list is used to demonstrate how different sorting algorithms work.
    After two passes through shuttle sort the resulting list is $$\begin{array} { l l l l l l l } 17 & 23 & 84 & 21 & 66 & 35 & 12 \end{array}$$
    1. How many different possibilities are there for the original list? Suppose, instead, that the same sort was carried out using bubble sort on the original list.
    2. Write down the list after two passes through bubble sort. The number of comparisons made is used as a measure of the run-time for a sorting algorithm.
    3. For a list of six values, what is the maximum total number of comparisons made in the first two passes of
      1. shuttle sort
      2. bubble sort? Steve used both shuttle sort and bubble sort on a list of five values. He says that shuttle sort is more efficient than bubble sort because it made fewer comparisons in the first two passes.
      3. Comment on what Steve said. The number of comparisons made when shuttle sort and bubble sort are used to sort every permutation of a list of four values is shown in the table below.
        Number of comparisons3456
        Shuttle sortNumber of permutations2688
        Bubble sortNumber of permutations10716
      4. Use the information in the table to decide which algorithm you would expect to have the quicker run-time. Justify your answer with calculations.
    OCR Further Discrete 2018 September Q3
    9 marks Challenging +1.2
    3 The pay-off matrix for a zero-sum game is
    XYZ
    \cline { 2 - 4 } A- 210
    \cline { 2 - 4 } B35- 3
    \cline { 2 - 4 } C- 4- 22
    \cline { 2 - 4 } D02- 1
    \cline { 2 - 4 }
    \cline { 2 - 4 }
    1. Show that the game does not have a stable solution.
    2. Use a graphical technique to find the optimal mixed strategy for the player on columns.
    3. Formulate an initial simplex tableau for the problem of finding the optimal mixed strategy for the player on rows.
    OCR Further Discrete 2018 September Q4
    19 marks Moderate -0.3
    4 A project is represented by the activity network below. The times are in days. \includegraphics[max width=\textwidth, alt={}, center]{22571082-016b-409b-bfeb-e7ebf48ccac7-4_384_935_1110_566}
    1. Explain the reason for each dummy activity.
    2. Calculate the early and late event times.
    3. Identify the critical activities.
    4. Calculate the independent float and interfering float on activity A .
    5. (a) Draw a cascade chart to represent the project, using the grid in the Printed Answer Booklet.
      (b) Describe the effect on
      The number of workers needed for each activity is shown below.
      ActivityABCDEFGH
      Workers21121111
      The project needs to be completed in at most 3 weeks ( 21 days).
      The duration of activity D is 9 days.
    6. Find the minimum number of workers needed. You should explain your reasoning carefully.
    OCR Further Discrete 2018 September Q5
    10 marks Moderate -0.8
    5 Consider the problem given below: $$\begin{array} { l l } \text { Minimise } & 4 \mathrm { AB } + 7 \mathrm { AC } + 8 \mathrm { BD } + 5 \mathrm { CD } + 5 \mathrm { CE } + 6 \mathrm { DF } + 3 \mathrm { EF } \\ \text { subject to } & \mathrm { AB } , \mathrm { AC } , \mathrm { BD } , \mathrm { CD } , \mathrm { CE } , \mathrm { DF } \text { and } \mathrm { EF } \text { are each either } 0 \text { or } 1 \\ & \mathrm { AB } + \mathrm { AC } + \mathrm { BD } + \mathrm { CD } + \mathrm { CE } + \mathrm { DF } + \mathrm { EF } = 5 \\ & \mathrm { AB } + \mathrm { AC } \geqslant 1 , \quad \mathrm { AB } + \mathrm { BD } \geqslant 1 , \quad \mathrm { AC } + \mathrm { CD } + \mathrm { CE } \geqslant 1 , \\ & \mathrm { BD } + \mathrm { CD } + \mathrm { DF } \geqslant 1 , \quad \mathrm { CE } + \mathrm { EF } \geqslant 1 , \quad \mathrm { DF } + \mathrm { EF } \geqslant 1 \end{array}$$
    1. Explain why this is not a standard LP formulation that could be set up as a Simplex tabulation. The variables \(\mathrm { AB } , \mathrm { AC } , \ldots\) correspond to arcs in a network. The weight on each arc is the coefficient of the corresponding variable in the objective function.
    2. Draw the network on the vertices in the Printed Answer Booklet. A variable that takes the value 1 corresponds to an arc that is used in the solution and a variable with the value 0 corresponds to an arc that is not used in the solution.
    3. Explain what is ensured by the constraint \(\mathrm { AB } + \mathrm { AC } \geqslant 1\). Julie claims that the solution to the problem will give the minimum spanning tree for the network.
    4. Find the minimum spanning tree for the network.
      Kim has a different network, exactly one of the arcs in this network is a directed arc.
      Kim wants to find a minimum weight set of arcs such that it is possible to get from any vertex to any other vertex.
    5. Explain why, if Kim's problem has a solution, the directed arc cannot be part of it.
    OCR Further Discrete 2018 September Q6
    8 marks Standard +0.3
    6 Kai mixes hot drinks using coffee and steamed milk.
    The amounts ( ml ) needed and profit ( \(\pounds\) ) for a standard sized cup of four different drinks are given in the table. The table also shows the amount of the ingredients available.
    Type of drinkCoffeeFoamed milkProfit
    w Americano8001.20
    \(x\) Cappuccino60120X
    \(y\) Flat White601001.40
    \(z\) Latte401201.50
    Available9001500
    Kai makes the equivalent of \(w\) standard sized americanos, \(x\) standard sized cappuccinos, \(y\) standard sized flat whites and \(z\) standard sized lattes. He can make different sized drinks so \(w , x , y , z\) need not be integers. Kai wants to find the maximum profit that he can make, assuming that the customers want to buy the drinks he has made.
    1. What is the minimum value of X for it to be worthwhile for Kai to make cappuccinos? Kai makes no cappuccinos.
    2. Use the simplex algorithm to solve Kai's problem. The grids in the Printed Answer Booklet should have at least enough rows and columns and there should be at least enough grids to show all the iterations needed. Only record the output from each iteration, not any intermediate stages.
      Interpret the solution and state the maximum profit that Kai can make.
    OCR Further Discrete 2018 September Q7
    13 marks Challenging +1.8
    7 A simply connected graph has 6 vertices and 10 arcs.
    1. What is the maximum vertex degree? You are now given that the graph is also Eulerian.
    2. Explaining your reasoning carefully, show that exactly two of the vertices have degree 2 .
    3. Prove that the vertices of degree 2 cannot be adjacent to one another.
    4. Use Kuratowski's theorem to show that the graph is planar.
    5. Show that it is possible to make a non-planar graph by the addition of one more arc. A digraph is created from a simply connected graph with 6 vertices and \(10 \operatorname { arcs }\) by making each arc into a single directed arc.
    6. What can be deduced about the indegrees and outdegrees?
    7. If a Hamiltonian cycle exists on the digraph, what can be deduced about the indegrees and outdegrees? \section*{OCR} \section*{Oxford Cambridge and RSA}
    OCR Further Pure Core 1 2018 December Q1
    5 marks Standard +0.3
    1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.
    1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
    2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).
    OCR Further Pure Core 1 2018 December Q2
    9 marks Standard +0.8
    2 The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8315a796-0e7d-464f-8604-9fe3ab7af359-2_470_657_913_319}
    1. The greatest value of \(r\) on the curve occurs at the point \(P\).
      1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
      2. Find the value of \(r\) at the point \(P\).
      3. Mark the point \(P\) on the copy of the graph in the Printed Answer Booklet.
    2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.
    OCR Further Pure Core 1 2018 December Q3
    7 marks Standard +0.3
    3 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).
    1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
    2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
    3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).
    OCR Further Pure Core 1 2018 December Q4
    4 marks Standard +0.3
    4 In this question you must show detailed reasoning.
    You are given that \(z = \sqrt { 3 } + \mathrm { i }\). \(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.
    1. Determine the value of \(n\).
    2. Find the value of \(z ^ { n }\).
    OCR Further Pure Core 1 2018 December Q5
    6 marks Standard +0.3
    5 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1 \\ - 8 & 4 & 0 \\ 19 & - 8 & - 1 \end{array} \right)\).
    1. Find \(\mathbf { A B }\).
    2. Hence write down \(\mathbf { A } ^ { - 1 }\).
    3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0 \\ 2 x + 5 y + 2 z = 1 \\ 3 x - 2 y - z = 4 \end{array}$$
      1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
      2. Find this unique solution.
    OCR Further Pure Core 1 2018 December Q6
    5 marks Moderate -0.3
    6 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .