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OCR Further Pure Core AS 2020 November Q4
6 marks Standard +0.3
4 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1 \\ x + b ^ { 2 } y = 3 \end{array}$$ where \(a\) and \(b\) are real numbers.
  1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\).
  2. Explain why the method used in part (a) works for all values of \(a\) and \(b\).
OCR Further Pure Core AS 2020 November Q6
5 marks Standard +0.8
6 Prove that \(n ! > 2 ^ { 2 n }\) for all integers \(n \geqslant 9\).
OCR Further Pure Core AS 2020 November Q7
6 marks Standard +0.3
7 The equations of two intersecting lines are \(\mathbf { r } = \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) \quad \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } - 3 \\ 1 \\ - 1 \end{array} \right)\) where \(a\) is a constant.
  1. Find a vector, \(\mathbf { b }\), which is perpendicular to both lines.
  2. Show that \(\mathbf { b } \cdot \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) = \mathbf { b } \cdot \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right)\).
  3. Hence, or otherwise, find the value of \(a\).
OCR Further Pure Core AS 2020 November Q8
8 marks Challenging +1.2
8 Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\} \\ & C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 i ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ where \(d\) is a real number.
  1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
    [0pt] [You may assume that \(C _ { 1 } \cap C _ { 2 } \neq \varnothing\).]
  2. Explain why the solution found in part (a) is not valid when \(d = 3\). \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Further Pure Core AS Specimen Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
The equation \(x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\). The equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
Find the values of \(p\) and \(q\).
OCR Further Pure Core AS Specimen Q2
4 marks Moderate -0.5
2 In this question you must show detailed reasoning.
Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = - 1 - \mathrm { i }\), find the following, giving each in the form \(a + b \mathrm { i }\).
  1. \(z _ { 1 } ^ { * } z _ { 2 }\)
  2. \(\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }\)
OCR Further Pure Core AS Specimen Q3
9 marks Moderate -0.8
3
  1. You are given two matrices, A and B, where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2 \\ 2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
  2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5 \\ 1 & 1 & 3 \\ - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2 \\ - 5 & 9 & - 1 \\ 3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
  3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } \\ 3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).
OCR Further Pure Core AS Specimen Q4
4 marks Standard +0.3
4 Draw the region of the Argand diagram for which \(| z - 3 - 4 i | \leq 5\) and \(| z | \leq | z - 2 |\).
OCR Further Pure Core AS Specimen Q5
9 marks Standard +0.3
5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\).
  1. The diagram in the Printed Answer Booklet shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
  2. Find the equation of the line of invariant points of this transformation.
  3. (a) Find the determinant of \(\mathbf { M }\).
    (b) Describe briefly how this value relates to the transformation represented by \(\mathbf { M }\).
OCR Further Pure Core AS Specimen Q6
6 marks Standard +0.3
6 At the beginning of the year John had a total of \(\pounds 2000\) in three different accounts. He has twice as much money in the current account as in the savings account.
  • The current account has an interest rate of \(2.5 \%\) per annum.
  • The savings account has an interest rate of \(3.7 \%\) per annum.
  • The supersaver account has an interest rate of \(4.9 \%\) per annum.
John has predicted that he will earn a total interest of \(\pounds 92\) by the end of the year.
  1. Model this situation as a matrix equation.
  2. Find the amount that John had in each account at the beginning of the year.
  3. In fact, the interest John will receive is \(\pounds 92\) to the nearest pound. Explain how this affects the calculations.
OCR Further Pure Core AS Specimen Q7
9 marks Challenging +1.2
7 In this question you must show detailed reasoning.
It is given that \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } - 13 z ^ { 2 } + 65 z - 125\).
The points representing the three roots of the equation \(\mathrm { f } ( z ) = 0\) are plotted on an Argand diagram.
Show that these points lie on the circle \(| z | = k\), where \(k\) is a real number to be determined.
OCR Further Pure Core AS Specimen Q8
5 marks Standard +0.3
8 Prove that \(n ! > 2 ^ { n }\) for \(n \geq 4\).
OCR Further Pure Core AS Specimen Q9
11 marks Standard +0.3
9
  1. Find the value of \(k\) such that \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { r } - 2 \\ 3 \\ k \end{array} \right)\) are perpendicular. Two lines have equations \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 3 \\ 2 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 5 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)\).
  2. Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  3. The vector \(\left( \begin{array} { l } 1 \\ a \\ b \end{array} \right)\) is perpendicular to the lines \(l _ { 1 }\) and \(l _ { 2 }\). Find the values of \(a\) and \(b\). \section*{END OF QUESTION PAPER} \section*{Copyright Information:} 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. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
OCR Further Statistics AS 2018 June Q1
8 marks Moderate -0.8
1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the Xth weekday of September.
  1. State an assumption needed for \(X\) to be well modelled by a geometric distribution.
  2. Find \(\mathrm { P } ( X = 11 )\).
  3. Find \(\mathrm { P } ( X \leqslant 8 )\).
  4. Find \(\operatorname { Var } ( X )\).
  5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.
OCR Further Statistics AS 2018 June Q2
8 marks Moderate -0.5
2 The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(\mathrm { P } ( W = w )\)0.250.36\(x\)\(x ^ { 2 }\)
  1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
  2. Find \(\operatorname { Var } ( 3 W + 6 )\).
OCR Further Statistics AS 2018 June Q3
6 marks Moderate -0.8
3 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).
  1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
  2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
  3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4.
OCR Further Statistics AS 2018 June Q4
8 marks Standard +0.3
4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$
  1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
  2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
  3. Assume now that the assumption in part (ii) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
  4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
OCR Further Statistics AS 2018 June Q5
8 marks Standard +0.3
5
  1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
    Find the probability that the team contains no more than 3 girls.
  2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).
OCR Further Statistics AS 2018 June Q7
8 marks Moderate -0.8
7 An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.
\(c\)1.941.781.621.511.521.4
\(m\)6.57.27.47.68.39.7
  1. State which, if either, of \(m\) and \(c\) is an independent variable.
  2. Calculate the equation of the least squares regression line of \(c\) on \(m\).
  3. State what effect, if any, there would be on your answer to part (ii) if the masses of the fish had been recorded in kilograms rather than pounds. ( \(1 \mathrm {~kg} \approx 2.2\) pounds.)
  4. The data is illustrated in the scatter diagram. Explain what is meant by 'least squares', illustrating your answer using the copy of this diagram in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{708e125e-43a8-40d8-94db-0ed80337d273-4_719_1043_961_513}
OCR Further Statistics AS 2018 June Q8
9 marks Challenging +1.2
8 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\).
Range\(1 \leqslant x \leqslant 8\)\(9 \leqslant x \leqslant 12\)\(13 \leqslant x \leqslant 20\)
Observed frequency12\(y\)\(28 - y\)
Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}
OCR Further Statistics AS 2019 June Q1
9 marks Moderate -0.8
1 When a spinner is spun, the outcome is equally likely to be 1,2 or 3 . In a competition, the spinner is spun twice and the outcomes are added to give a total score \(T\).
  1. Show that the expectation of \(T\) is 4 .
  2. Find the variance of \(T\). A competitor pays \(\pounds 1.50\) to enter the competition and receives \(\pounds X\), where \(X = 0.3 T\).
    1. Find the expectation of the competitor's profit.
    2. Find the variance of the competitor's profit.
OCR Further Statistics AS 2019 June Q2
6 marks Standard +0.3
2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).
  1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
  2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
  3. State a necessary assumption for the validity of your calculation in part (b).
OCR Further Statistics AS 2019 June Q3
6 marks Challenging +1.2
3
  1. Shula calculates the value of Spearman's rank correlation coefficient \(r _ { s }\) for 9 pairs of rankings.
    Find the largest possible value of \(r _ { s }\) that Shula can obtain that is less than 1 .
  2. A set of bivariate data consists of 5 pairs of values. It is known that for this data the value of Spearman's rank correlation coefficient is - 1 but the value of Pearson's product-moment correlation coefficient is not - 1 . Sketch a possible scatter diagram illustrating the data.
OCR Further Statistics AS 2019 June Q4
7 marks Moderate -0.3
4 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.
  1. Find the probability that all the men are next to each other.
  2. Find the probability that no two men are next to one another.
OCR Further Statistics AS 2019 June Q5
7 marks Standard +0.3
5 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.
  1. For all sixteen candidates, the value of the product moment correlation coefficient \(r\) for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the \(5 \%\) significance level, of association between the marks on the two papers.
  2. A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by \(n = 10 , \sum x = 750 , \sum y = 690 , \sum x ^ { 2 } = 57690 , \sum y ^ { 2 } = 49676 , \sum x y = 50829\).
    1. Calculate the value of \(r\) for these 10 candidates.
    2. What do the two values of \(r\), in parts (a) and (b)(i), tell you about the scores of the sixteen candidates?