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OCR MEI Further Statistics B AS 2021 November Q5
9 marks Standard +0.3
5 A food company makes mini apple pies. The weight of pastry in a pie is Normally distributed with mean 75 g and standard deviation 4 g . The weight of filling in a pie is Normally distributed with mean 130 g and standard deviation 8 g . You should assume that the weights of pastry and filling in a pie are independent.
  1. Find the probability that the weight of pastry in a randomly chosen pie is between 70 g and 80 g .
  2. Find the probability that the mean weight of filling in 10 randomly chosen pies is at least 125 g. The pies are sold in packs of 4 . The weight of the packaging is Normally distributed with mean 165 g and standard deviation 6 g .
  3. In order to find the probability that the total weight of a pack of 4 pies is less than 1 kg , you must assume that the weight of the packaging is independent of the weight of the pies.
    1. State another necessary assumption.
    2. Given that the assumptions are valid, calculate this probability.
OCR MEI Further Statistics B AS 2021 November Q6
11 marks Standard +0.3
6 The probability density function of the continuous random variable \(X\) is given by \(f ( x ) = \begin{cases} 2 ( 1 + a x ) & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise } , \end{cases}\) where \(a\) is a constant.
  1. Show that \(a = - 1\).
  2. Find the cumulative distribution function of \(X\).
  3. Find \(\mathrm { P } ( X < 0.5 )\).
  4. Show that \(\mathrm { E } ( X )\) is greater than the median of \(X\).
OCR MEI Further Statistics B AS Specimen Q1
9 marks Easy -1.2
1 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins \(\pounds 5\). Otherwise the player wins nothing. Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them. \begin{table}[h]
C5\(\times \vee f _ { x }\)=RANDBETWEEN(1,6)
ABCDEFGHJ
1dice 1dice 2dice 3dice 4dice 5dice 6High scoreLow scoreDifference
2game 1224233422
3game 2263212615
4game 3315346615
5game 4652563624
6game 5633532624
7game 6563514615
8game 7231264615
9game 8666615615
10game 9362541615
11game 10511461615
12game 11256165615
13game 12256666624
14game 13222244422
15game 14166635615
16game 15223351514
17game 16123433413
18game 17524216615
19game 18615215615
20game 19135135514
21game 20543251
\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table}
  1. (A) Write down the numbers in columns H , I and J for game 20 .
    (B) Use the spreadsheet to estimate the probability of a player winning a game.
  2. State how the estimate of probability in (i) (B) could be improved.
  3. Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability. All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
  4. If Abby charges \(\pounds 1\) per game, estimate the total profit when 50 players each play the game once.
OCR MEI Further Statistics B AS Specimen Q2
7 marks Standard +0.3
2 The cumulative distribution function of the continuous random variable, \(Y\), is given below. $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { y ^ { 3 } - y ^ { 2 } } { 4 } & 1 \leq y \leq 2 \\ 1 & y > 2 \end{array} \right.$$
  1. Find \(\mathrm { P } ( Y \leq 1.5 )\)
  2. Verify that the median of \(Y\) lies between 1.6 and 1.7.
  3. Find the probability density function of \(Y\).
OCR MEI Further Statistics B AS Specimen Q3
11 marks Standard +0.3
3 At a factory, flour is packed into bags. A model for the mass in grams of flour packed into each bag is \(1500 + X\), where \(X\) is a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x ( 6 - x ) & 0 \leq x \leq 6 \\ 0 & \text { elsewhere, } \end{array} \right.$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 36 }\).
  2. Find the probability that a randomly selected bag of flour contains 1505 grams of flour or more.
  3. Find
    • the mean of \(X\),
    • the standard deviation of \(X\).
OCR MEI Further Statistics B AS Specimen Q4
8 marks Standard +0.3
4 An online encyclopedia claims that the average mass of an adult European hedgehog is 720 g . In an investigation to check this average figure, the masses in grams of twelve randomly chosen adult European hedgehogs are measured and shown below.
705730720691718680
731723745708724736
  1. What assumption is required to carry out a Wilcoxon test in this situation?
  2. Given that this assumption is met, carry out a 2 -tail Wilcoxon test at the \(5 \%\) level to test whether the median mass is 720 g . You should state your hypotheses and complete the table of calculations in the Printed Answer Booklet.
OCR MEI Further Statistics B AS Specimen Q5
11 marks Standard +0.3
5 A particular alloy of bronze is specified as containing \(11.5 \%\) copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows.
11.1211.2911.4211.4311.2011.2511.65
11.3311.5611.3411.4411.2411.6011.52
The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-4_428_1550_1434_299} \captionsetup{labelformat=empty} \caption{Fig 5.1}
\end{figure}
  1. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data. The researcher uses software to produce a \(99 \%\) confidence interval for the mean percentage of copper in the alloy, based on the \(t\) distribution. The output from the software is shown in Fig 5.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726} \captionsetup{labelformat=empty} \caption{Fig 5.2}
    \end{figure}
  2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
  3. (A) State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
    (B) State whether the assumption in part (iii) (A) seems reasonable.
  4. Does the confidence interval suggest that the copper content is different from \(11.5 \%\), on average? Explain your answer.
  5. In the output from the software shown in Fig 5.2, SE stands for 'standard error'.
    (A) Explain what a standard error is.
    (B) Show how the standard error was calculated in this case.
  6. Suggest a way in which the researcher could produce a narrower confidence interval.
OCR MEI Further Statistics B AS Specimen Q6
8 marks Standard +0.3
6 The table below shows the mean and variance of the test scores of a random samples of 70 girls who are starting an A level Mathematics course.
Sample meanSample variance
118.8686.57
  1. Showing your working, find a \(95 \%\) confidence interval for the population mean.
  2. Explain why you can construct the interval in part (i) despite no information about the distribution of the parent population being given.
  3. The same random sample of girls repeats the test. The mean improvement in score is 0.9 . The \(95 \%\) confidence interval for the improvement is \([ - 1.5,3.3 ]\). What is the sample variance for the improvement in score?
OCR MEI Further Statistics B AS Specimen Q7
6 marks Standard +0.3
7 Two flatmates work at the same location. One of them takes the bus to work and the other one cycles. Journey times, measured in minutes, are distributed as follows.
  • By bus: Normally distributed with mean 23 and standard deviation 6
  • By bicycle: Normally distributed with mean 21 and standard deviation 2
You should assume that all journey times are independent.
  1. One morning the two flatmates set out at the same time. Find the probability that the person who takes the bus arrives before the cyclist.
  2. Find the probability that the total time taken for 5 bus journeys is less than 2 hours.
  3. Comment on the assumption that all journey times are independent. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{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.
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OCR MEI Further Pure Core 2019 June Q1
4 marks Easy -1.2
1 Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.
OCR MEI Further Pure Core 2019 June Q2
3 marks Moderate -0.5
2 The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
OCR MEI Further Pure Core 2019 June Q3
7 marks Moderate -0.8
3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } k & 1 \\ 2 & 0 \end{array} \right)\), where \(k\) is a constant.
  1. Verify the result \(( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) in this case.
  2. Investigate whether \(\mathbf { A }\) and \(\mathbf { B }\) are commutative under matrix multiplication.
OCR MEI Further Pure Core 2019 June Q5
5 marks Standard +0.3
5 Using the Maclaurin series for \(\cos 2 x\), show that, for small values of \(x\), \(\sin ^ { 2 } x \approx a x ^ { 2 } + b x ^ { 4 } + c x ^ { 6 }\),
where the values of \(a , b\) and \(c\) are to be given in exact form.
OCR MEI Further Pure Core 2019 June Q6
4 marks Standard +0.8
6 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).
OCR MEI Further Pure Core 2019 June Q7
8 marks Standard +0.3
7 A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y\), where \(c\) is a positive constant.
  1. Show that the polar equation of the curve is \(r ^ { 2 } = c ^ { 2 } \sin 2 \theta\).
  2. Sketch the curves \(r = c \sqrt { \sin 2 \theta }\) and \(r = - c \sqrt { \sin 2 \theta }\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  3. Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
    Answer all the questions.
OCR MEI Further Pure Core 2019 June Q9
7 marks Moderate -0.3
9 Prove by induction that \(5 ^ { n } + 2 \times 11 ^ { n }\) is divisible by 3 for all positive integers \(n\).
OCR MEI Further Pure Core 2019 June Q11
12 marks Standard +0.3
11
  1. Specify fully the transformations represented by the following matrices.
    • \(\mathbf { M } _ { 1 } = \left( \begin{array} { r r } \frac { 3 } { 5 } & - \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\)
    • \(\mathbf { M } _ { 2 } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\)
    • Find the equation of the mirror line of the reflection R represented by the matrix \(\mathbf { M } _ { 3 } = \mathbf { M } _ { 1 } \mathbf { M } _ { 2 }\).
    • It is claimed that the reflection represented by the matrix \(\mathbf { M } _ { 4 } = \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }\) has the same mirror line as R . Explain whether or not this claim is correct.
OCR MEI Further Pure Core 2019 June Q12
9 marks Challenging +1.2
12 Three intersecting lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have equations \(L _ { 1 } : \frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 1 } , \quad L _ { 2 } : \frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { - 4 } \quad\) and \(\quad L _ { 3 } : \frac { x - 1 } { 1 } = \frac { y - 2 } { 1 } = \frac { z + 4 } { 5 }\).
Find the area of the triangle enclosed by these lines.
OCR MEI Further Pure Core 2019 June Q13
11 marks Standard +0.8
13
  1. Using the logarithmic form of \(\operatorname { arcosh } x\), prove that the derivative of \(\operatorname { arcosh } x\) is \(\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
  2. Hence find \(\int _ { 1 } ^ { 2 } \operatorname { arcosh } x \mathrm {~d} x\), giving your answer in exact logarithmic form.
  3. Ali tries to evaluate \(\int _ { 0 } ^ { 1 } \operatorname { arcosh } x \mathrm {~d} x\) using his calculator, and gets an 'error'. Explain why.
OCR MEI Further Pure Core 2019 June Q14
13 marks Challenging +1.2
14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.
  1. In the case where the planes do not intersect at a unique point,
    1. find \(b\) in terms of \(a\),
    2. find the value of \(c\) for which the planes form a sheaf.
  2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).
OCR MEI Further Pure Core 2019 June Q16
12 marks Challenging +1.2
16
  1. Show that \(\left( 2 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 2 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by \(C = \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \cos n \theta\), \(S = \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \sin n \theta\).
  2. Show that \(C = \frac { 2 ^ { n } ( 2 \cos \theta - 1 ) - 2 \cos ( n + 1 ) \theta + \cos n \theta } { 2 ^ { n } ( 5 - 4 \cos \theta ) }\).
OCR MEI Further Pure Core 2019 June Q17
22 marks
17 A cyclist accelerates from rest for 5 seconds then brakes for 5 seconds, coming to rest at the end of the 10 seconds. The total mass of the cycle and rider is \(m \mathrm {~kg}\), and at time \(t\) seconds, for \(0 \leqslant t \leqslant 10\), the cyclist's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance to motion, modelled by a force of magnitude 0.1 mvN , acts on the cyclist during the whole 10 seconds.
  1. Explain why modelling the resistance to motion in this way is likely to be more realistic than assuming this force is constant. During the braking phase of the motion, for \(5 \leqslant t \leqslant 10\), the brakes apply an additional constant resistance force of magnitude \(2 m \mathrm {~N}\) and the cyclist does not provide any driving force.
  2. Show that, for \(5 \leqslant t \leqslant 10 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = - 2\).
    1. Solve the differential equation in part (b).
    2. Hence find the velocity of the cyclist when \(t = 5\). During the acceleration phase ( \(0 \leqslant t \leqslant 5\) ), the cyclist applies a driving force of magnitude directly proportional to \(t\).
  4. Show that, for \(0 \leqslant t \leqslant 5 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = \lambda t\), where \(\lambda\) is a positive constant.
    1. Show by integration that, for \(0 \leqslant t \leqslant 5 , v = 10 \lambda \left( t - 10 + 10 \mathrm { e } ^ { - 0.1 t } \right)\).
    2. Hence find \(\lambda\).
  6. Find the total distance, to the nearest metre, travelled by the cyclist during the motion.
OCR MEI Further Pure Core 2022 June Q1
7 marks Standard +0.3
1
  1. By considering \(( r + 1 ) ^ { 3 } - r ^ { 3 }\), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 3 \mathrm { r } ^ { 2 } + 3 \mathrm { r } + 1 \right)\).
  2. Use this result to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\), expressing your answer in fully factorised form.
OCR MEI Further Pure Core 2022 June Q2
5 marks Challenging +1.2
2 In this question you must show detailed reasoning. Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x\)
OCR MEI Further Pure Core 2022 June Q3
6 marks Standard +0.3
3 In this question you must show detailed reasoning.
Solve the equation \(3 \cosh x = 2 \sinh ^ { 2 } x\), giving your solutions in exact logarithmic form.