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OCR Further Pure Core 2 Specimen Q1
4 marks Moderate -0.3
1 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 )\). Give your answer in a fully factorised form.
OCR Further Pure Core 2 Specimen Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
  1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
  2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.
OCR Further Pure Core 2 Specimen Q4
5 marks Challenging +1.2
4 It is given that \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + k x } \equiv \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + k }\) where \(k , A , B\) and \(C\) are positive integers.
Determine the set of possible values of \(k\).
OCR Further Pure Core 2 Specimen Q5
4 marks Standard +0.8
5 In this question you must show detailed reasoning.
Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]
OCR Further Pure Core 2 Specimen Q6
8 marks Standard +0.8
6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).
  1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
  2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).
OCR Further Pure Core 2 Specimen Q8
8 marks Standard +0.8
8 The equation of a curve is \(y = \cosh ^ { 2 } x - 3 \sinh x\). Show that \(\left( \ln \left( \frac { 3 + \sqrt { 13 } } { 2 } \right) , - \frac { 5 } { 4 } \right)\) is the only stationary point on the curve.
OCR Further Pure Core 2 Specimen Q9
6 marks Standard +0.8
9 A curve has equation \(x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }\), where \(x\) and \(y\) are not both zero.
  1. Show that the equation of the curve in polar coordinates is \(r ^ { 2 } = \frac { 2 } { 2 - \sin ^ { 2 } 2 \theta }\).
  2. Deduce that no point on the curve \(x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }\) is further than \(\sqrt { 2 }\) from the origin.
OCR Further Pure Core 2 Specimen Q10
8 marks Challenging +1.2
10 Let \(C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos r \theta\). Show that \(C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos 10 \theta\).
OCR Further Pure Core 2 Specimen Q11
17 marks Challenging +1.2
11 During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 0.3 x - 0.2 y \text { and } \frac { \mathrm { d } z } { \mathrm {~d} t } = 0.2 y + 0.1 x$$ where \(x , y\) and \(z\) are the amounts in kg of \(X , Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substances \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = A \mathrm { e } ^ { - 0.4 t }\), stating the value of \(A\).
  2. (a) Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } z } { \mathrm {~d} t } = 0\).
    (b) Comment on this result in the context of the industrial process.
  3. Express \(y\) in terms of \(t\).
  4. Determine the maximum amount of substance \(Y\) present during the process.
  5. How long does it take to produce 9 kg of substance \(Z\) ? \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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OCR Further Statistics 2019 June Q1
5 marks Standard +0.3
1 A set of bivariate data ( \(X , Y\) ) is summarised as follows. \(n = 25 , \sum x = 9.975 , \sum y = 11.175 , \sum x ^ { 2 } = 5.725 , \sum y ^ { 2 } = 46.200 , \sum x y = 11.575\)
  1. Calculate the value of Pearson's product-moment correlation coefficient.
  2. Calculate the equation of the regression line of \(y\) on \(x\). It is desired to know whether the regression line of \(y\) on \(x\) will provide a reliable estimate of \(y\) when \(x = 0.75\).
  3. State one reason for believing that the estimate will be reliable.
  4. State what further information is needed in order to determine whether the estimate is reliable.
OCR Further Statistics 2019 June Q2
4 marks Standard +0.3
2 The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.
  1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520.
  2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval.
OCR Further Statistics 2019 June Q3
6 marks Challenging +1.2
3 Six red counters and four blue counters are arranged in a straight line in a random order.
Find the probability that
  1. no blue counter has fewer than two red counters between it and the nearest other blue counter,
  2. no two blue counters are next to one another.
OCR Further Statistics 2019 June Q4
9 marks Standard +0.3
4 The greatest weight \(W N\) that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80 . A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N .
  1. Test at the \(1 \%\) significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support.
  2. State an assumption needed in carrying out the test in part (a).
  3. Explain whether it is necessary to use the central limit theorem in carrying out the test.
OCR Further Statistics 2019 June Q5
7 marks Standard +0.8
5 Five runners, \(A , B , C , D\) and \(E\), take part in two different races.
Spearman’s rank correlation coefficient for the orders in which the runners finish is calculated and a test for positive agreement is carried out at the \(5 \%\) significance level.
  1. State suitable hypotheses for the test.
  2. Find the largest possible value of \(\sum d ^ { 2 }\) for which the result of the test is to reject the null hypothesis.
  3. In the first race, the order in which the five runners finished was: \(A , B , C , D , E\). In the second race, three of the runners finished in the same positions as in the first race. The result of the test is to reject the null hypothesis. Find a possible order for the runners to finish in the second race.
OCR Further Statistics 2019 June Q6
10 marks Standard +0.3
6 Yusha investigates the proportion of left-handed people living in two cities, \(A\) and \(B\). He obtains data from random samples from the two cities. His results are shown in the table, in which \(L\) denotes "left-handed".
\(L\)\(L ^ { \prime }\)
\(A\)149
\(B\)2651
  1. Test at the 10\% significance level whether there is association between being left-handed and living in a particular city. A person is chosen at random from one of the cities \(A\) and \(B\).
    Let \(A\) denote "the person lives in city \(A\) ".
  2. State the relationship between \(\mathrm { P } ( L )\) and \(P ( L \mid A )\) according to the model implied by the null hypothesis of your test.
  3. Use the data in the table to suggest a value for \(P ( L \mid A )\) given by an improved model.
OCR Further Statistics 2019 June Q7
10 marks Standard +0.3
7 The random variable \(D\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\operatorname { Var } ( D ) = \frac { 40 } { 9 }\).
Determine
  1. \(\operatorname { Var } ( 3 D + 5 )\),
  2. \(\mathrm { E } ( 3 \mathrm { D } + 5 )\),
  3. \(\mathrm { P } ( D > \mathrm { E } ( D ) )\).
OCR Further Statistics 2019 June Q8
10 marks Standard +0.3
8 A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726 . Test at the 5\% significance level whether there is a difference in the ranks of the students taught by the two professors.
OCR Further Statistics 2019 June Q9
14 marks Standard +0.8
9 The continuous random variable \(T\) has cumulative distribution function \(F ( t ) = \begin{cases} 0 & t < 0 , \\ 1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}\)
  1. Find the cumulative distribution function of \(2 T\).
  2. Show that, for constant \(k , \mathrm { E } \left( \mathrm { e } ^ { k t } \right) = \frac { 1 } { 1 - 4 k }\). You should state with a reason the range of values of \(k\) for which this result is valid.
  3. \(\quad T\) is the time before a certain event occurs. Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\). \section*{END OF QUESTION PAPER}
OCR Further Statistics 2022 June Q1
5 marks Moderate -0.3
1 A researcher wishes to find people who say that they support a specific plan. Each day the researcher interviews people at random, one after the other, until they find one person who says that they support this plan. The researcher does not then interview any more people that day. The total number of people interviewed on any one day is denoted by \(R\).
  1. Assume that in fact \(1 \%\) of the population would say that they support the plan.
    1. State an appropriate distribution with which to model \(R\), giving the value(s) of any parameter(s).
    2. Find \(\mathrm { P } ( 50 < R \leqslant 150 )\). The researcher incorrectly believes that the variance of a random variable \(X\) with any discrete probability distribution is given by the formula \([ \mathrm { E } ( X ) ] ^ { 2 } - \mathrm { E } ( X )\).
  2. Show that, for the type of distribution stated in part (a), they will obtain the correct value of the variance, regardless of the value(s) of the parameter(s).
OCR Further Statistics 2022 June Q2
11 marks Moderate -0.8
2 The directors of a large company believe that there are more computer failures in the Head Office when temperatures are higher. They obtain data for the Head Office for the maximum temperature, \(T ^ { \circ } \mathrm { C }\), and the number of computer failures, \(X\), on each of 12 randomly chosen days.
  1. State which of the following words can be applied to \(T\). Dependent Independent Controlled Response The data is summarised as follows. \(n = 12 \quad \sum t = 261 \quad \sum x = 41 \quad \sum t ^ { 2 } = 5869 \quad \sum x ^ { 2 } = 311 \quad \sum \mathrm { tx } = 1021\)
  2. Calculate the value of the product moment correlation coefficient \(r\).
  3. The directors wish to investigate their belief using a significance test at the \(1 \%\) level.
    1. Explain why a 1-tail test is appropriate in this situation.
    2. Carry out the test.
  4. One of the directors prefers the temperatures to be given in Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), rather than Centigrade ( \({ } ^ { \circ } \mathrm { C }\) ). The relationship between F and C is \(\mathrm { F } = \frac { 9 } { 5 } \mathrm { C } + 32\).
    State the value of \(r\) that would result from using temperatures in Fahrenheit in the calculation.
OCR Further Statistics 2022 June Q4
9 marks Moderate -0.8
4 The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.
  1. State a further assumption needed for the Poisson model to be valid.
  2. State the value of the standard deviation of \(R\).
  3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\operatorname { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
  4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
  5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.
OCR Further Statistics 2022 June Q5
10 marks Standard +0.3
5 A company uses two drivers for deliveries.
Driver \(A\) charges a fixed rate of \(\pounds 80\) per day plus \(\pounds 2\) per mile travelled on that day. Driver \(B\) charges a fixed rate of \(\pounds 120\) per day plus \(\pounds 1.50\) per mile travelled on that day.
On each working day the total distance, in miles, travelled by each driver is a random variable with the distribution \(\mathrm { N } ( 83,360 )\).
  1. Find the probability that driver \(A\) charges the company less than \(\pounds 235.00\) for a randomly chosen day’s deliveries.
  2. Find the probability that the total charge to the company of three randomly chosen days' deliveries by driver \(A\) is at least \(\pounds 300\) more than the total charge of two randomly chosen days' deliveries by driver \(B\).
OCR Further Statistics 2022 June Q6
7 marks Challenging +1.2
6 The random variable \(X\) was assumed to have a normal distribution with mean \(\mu\). Using a random sample of size 128, a significance test was carried out using the following hypotheses. \(\mathrm { H } _ { 0 } : \mu = 30\) \(\mathrm { H } _ { 1 } : \mu > 30\) It was found that \(\sum x = 3929.6\) and \(\sum x ^ { 2 } = 123483.52\). The conclusion of the test was to reject the null hypothesis.
  1. Determine the range of possible values of the significance level of the test.
  2. It was subsequently found that \(X\) was not normally distributed. Explain whether this invalidates the conclusion of the test.
OCR Further Statistics 2022 June Q7
8 marks
7 The continuous random variable \(X\) has probability density function \(f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant and \(n\) is a parameter whose value is positive. It is given that the median of \(X\) is 0.8816 correct to 4 decimal places. Ten independent observations of \(X\) are obtained. Find the expected number of observations that are less than 0.8 .
OCR Further Statistics 2022 June Q8
7 marks
8 The critical region for an \(r\) \% two-tailed Wilcoxon signed-rank test, based on a large sample of size \(n\), is \(\left\{ W _ { + } \leqslant 113 \right\} \cup \left\{ W _ { + } \geqslant 415 \right\}\).
  1. Show that \(n = 32\).
  2. Using a suitable approximation, determine the value of \(r\).