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OCR MEI Further Statistics Minor Specimen Q3
10 marks Standard +0.3
A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between \(0\) and \(10\) (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.
  1. Let \(X\) be the number of points gained after shopping once. Find
  2. Let \(Y\) be the number of points gained after shopping twice. Find
  3. Find the probability of the most likely number of points gained after shopping twice. Justify your answer. [4]
OCR MEI Further Statistics Minor Specimen Q4
8 marks Moderate -0.3
  1. State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day. [2]
Jane records the number of junk emails which she receives each day. During working hours (\(9\)am to \(5\)pm, Monday to Friday) the mean number of junk emails is \(7.4\) per day. Outside working hours (\(5\)pm to \(9\)am), the mean number of junk emails is \(0.3\) per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".
  1. Find the probability that the number of junk emails which she receives between \(9\)am and \(5\)pm on a Monday is
    1. exactly \(10\), [1]
    2. at least \(10\). [2]
    1. What assumption must you make to calculate the probability that the number of junk emails which she receives from \(9\)am Monday to \(9\)am Tuesday is at most \(20\)? [1]
    2. Find the probability. [2]
OCR MEI Further Statistics Minor Specimen Q5
10 marks Standard +0.3
Each contestant in a talent competition is given a score out of \(20\) by a judge. The organisers suspect that the judge's scores are associated with the age of the contestant. Table \(5.1\) and the scatter diagram in Fig. \(5.2\) show the scores and ages of a random sample of \(7\) contestants.
ContestantABCDEFG
Age6651392992214
Score1211151716189
Table 5.1 \includegraphics{figure_1} Fig. 5.2 Contestant G did not finish her performance, so it is decided to remove her data.
  1. Spearman's rank correlation coefficient between age and score, including all \(7\) contestants, is \(-0.25\). Explain why Spearman's rank correlation coefficient becomes more negative when the data for contestant G is removed. [1]
  2. Calculate Spearman's rank correlation coefficient for the \(6\) remaining contestants. [3]
  3. Using this value of Spearman's rank correlation coefficient, carry out a hypothesis test at the \(5\%\) level to investigate whether there is any association between age and score. [5]
  4. Briefly explain why it may be inappropriate to carry out a hypothesis test based on Pearson's product moment correlation coefficient using these data. [1]
OCR MEI Further Statistics Minor Specimen Q6
16 marks Standard +0.3
At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of \(40\) birds.
Number of recaptures, \(x\)012345678910
Frequency255910431010
  1. The sample mean of \(x\) is \(3.4\). Calculate the sample variance of \(x\). [2]
  2. Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. [1]
The screenshot below shows part of a spreadsheet for a \(\chi^2\) test to assess the goodness of fit of a Poisson model. The sample mean of \(3.4\) has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted. \includegraphics{figure_2}
  1. State the null and alternative hypotheses for the test. [1]
  2. Calculate the missing values in cells
  3. Complete the test at the \(10\%\) significance level. [5]
  4. The screenshot below shows part of a spreadsheet for a \(\chi^2\) test for a different species of bird. Find the value of the Poisson parameter used. \includegraphics{figure_3} [3]
OCR MEI Further Statistics Minor Specimen Q7
4 marks Moderate -0.5
A fair coin has \(+1\) written on the heads side and \(-1\) on the tails side. The coin is tossed \(100\) times. The sum of the numbers showing on the \(100\) tosses is the random variable \(Y\). Show that the variance of \(Y\) is \(100\). [4]
OCR MEI Further Extra Pure 2019 June Q1
5 marks Moderate -0.3
The matrix A is \(\begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix}\)
  1. Given that A represents a reflection, write down the eigenvalues of A. [1]
  2. Hence find the eigenvectors of A. [3]
  3. Write down the equation of the mirror line of the reflection represented by A. [1]
OCR MEI Further Extra Pure 2019 June Q2
11 marks Challenging +1.2
A surface \(S\) is defined by \(z = 4x^2 + 4y^2 - 4x + 8y + 11\).
  1. Show that the point P\((0.5, -1, 6)\) is the only stationary point on \(S\). [2]
    1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\). [2]
    2. By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\). [3]
  3. In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\). Find all possible values of \(a\) and \(c\). [4]
OCR MEI Further Extra Pure 2019 June Q3
8 marks Challenging +1.2
The matrix A is \(\begin{pmatrix} -1 & 2 & 4 \\ 0 & -1 & -25 \\ -3 & 5 & -1 \end{pmatrix}\). Use the Cayley-Hamilton theorem to find A\(^{-1}\). [8]
OCR MEI Further Extra Pure 2019 June Q4
8 marks Challenging +1.8
\(T\) is the set \(\{1, 2, 3, 4\}\). A binary operation \(\bullet\) is defined on \(T\) such that \(a \bullet a = 2\) for all \(a \in T\). It is given that \((T, \bullet)\) is a group.
  1. Deduce the identity element in \(T\), giving a reason for your answer. [2]
  2. Find the value of \(1 \bullet 3\), showing how the result is obtained. [3]
    1. Complete a group table for \((T, \bullet)\). [2]
    2. State with a reason whether the group is abelian. [1]
OCR MEI Further Extra Pure 2019 June Q5
15 marks Standard +0.8
A financial institution models the repayment of a loan to a client in the following way.
  • An amount, \(£C\), is loaned to the client at the start of the repayment period.
  • The amount owed \(n\) years after the start of the repayment period is \(£L_n\), so that \(L_0 = C\).
  • At the end of each year, interest of \(\alpha\%\) (\(\alpha > 0\)) of the amount owed at the start of that year is added to the amount owed.
  • Immediately after interest has been added to the amount owed a repayment of \(£R\) is made by the client.
  • Once \(L_n\) becomes negative the repayment is finished and the overpayment is refunded to the client.
  1. Show that during the repayment period, \(L_{n+1} = aL_n + b\), giving \(a\) and \(b\) in terms of \(\alpha\) and \(R\). [2]
  2. Find the solution of the recurrence relation \(L_{n+1} = aL_n + b\) with \(L_0 = C\), giving your solution in terms of \(a\), \(b\), \(C\) and \(n\). [5]
  3. Deduce from parts (a) and (b) that, for the repayment scheme to terminate, \(R > \frac{\alpha C}{100}\). [2]
A client takes out a £30000 loan at 8% interest and agrees to repay £3000 at the end of each year.
    1. Use an algebraic method to find the number of years it will take for the loan to be repaid. [3]
    2. Taking into account the refund of overpayment, find the total amount that the client repays over the lifetime of the loan. [3]
OCR MEI Further Extra Pure 2019 June Q6
13 marks Challenging +1.8
  1. Given that \(\sqrt{7}\) is an irrational number, prove that \(a^2 - 7b^2 \neq 0\) for all \(a, b \in \mathbb{Q}\) where \(a\) and \(b\) are not both 0. [2]
  2. A set \(G\) is defined by \(G = \{a + b\sqrt{7} : a, b \in \mathbb{Q}, a\) and \(b\) not both 0\(\}\). Prove that \(G\) is a group under multiplication. (You may assume that multiplication is associative.) [7]
  3. A subset \(H\) of \(G\) is defined by \(H = \{1 + c\sqrt{7} : c \in \mathbb{Q}\}\). Determine whether or not \(H\) is a subgroup of \((G, \times)\). [2]
  4. Using \((G, \times)\), prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false. [2]
OCR MEI Further Extra Pure 2021 November Q1
11 marks Challenging +1.2
In this question you must show detailed reasoning. A surface \(S\) is defined by \(z = \mathrm{f}(x, y)\) where \(\mathrm{f}(x, y) = x^3 + x^2 y - 2y^2\).
  1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = \mathrm{f}(2, y)\) giving the coordinates of any turning points and any points of intersection with the axes. [4]
  2. Find the stationary points on \(S\). [7]
OCR MEI Further Extra Pure 2021 November Q2
7 marks Challenging +1.2
\(G\) is a group of order 8.
  1. Explain why there is no subgroup of \(G\) of order 6. [1]
You are now given that \(G\) is a cyclic group with the following features: • \(e\) is the identity element of \(G\), • \(g\) is a generator of \(G\), • \(H\) is the subgroup of \(G\) of order 4.
  1. Write down the possible generators of \(H\). [2]
\(M\) is the group \((\{0, 1, 2, 3, 4, 5, 6, 7\}, +_8)\) where \(+_8\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).
  1. Specify all possible isomorphisms between \(M\) and \(G\). [4]
OCR MEI Further Extra Pure 2021 November Q3
14 marks Challenging +1.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 3 & 3 & 0 \\ 0 & 2 & 2 \\ 1 & 3 & 4 \end{pmatrix}\).
  1. Determine the characteristic equation of \(\mathbf{A}\). [3]
  2. Hence verify that the eigenvalues of \(\mathbf{A}\) are 1, 2 and 6. [1]
  3. For each eigenvalue of \(\mathbf{A}\) determine an associated eigenvector. [4]
  4. Use the results of parts (b) and (c) to find \(\mathbf{A}^n\) as a single matrix, where \(n\) is a positive integer. [6]
OCR MEI Further Extra Pure 2021 November Q4
14 marks Challenging +1.3
The sequence \(u_0, u_1, u_2, \ldots\) satisfies the recurrence relation \(u_{n+2} - 3u_{n+1} - 10u_n = 24n - 10\).
  1. Determine the general solution of the recurrence relation. [6]
  2. Hence determine the particular solution of the recurrence relation for which \(u_0 = 6\) and \(u_1 = 10\). [3]
  3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u_2\). [1]
The sequence \(v_0, v_1, v_2, \ldots\) is defined by \(v_n = \frac{u_n}{p^n}\) for some constant \(p\), where \(u_n\) denotes the particular solution found in part (b). You are given that \(v_n\) converges to a finite non-zero limit, \(q\), as \(n \to \infty\).
  1. Determine \(p\) and \(q\). [4]
OCR MEI Further Extra Pure 2021 November Q5
6 marks Challenging +1.8
A surface \(S\) is defined for \(z \geqslant 0\) by \(x^2 + y^2 + 2z^2 = 126\). \(C\) is the set of points on \(S\) for which the tangent plane to \(S\) at that point intersects the \(x\)-\(y\) plane at an angle of \(\frac{1}{4}\pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined. [6]
OCR MEI Further Extra Pure 2021 November Q6
8 marks Challenging +1.8
You are given that \(q \in \mathbb{Z}\) with \(q \geqslant 1\) and that $$S = \frac{1}{(q+1)} + \frac{1}{(q+1)(q+2)} + \frac{1}{(q+1)(q+2)(q+3)} + \cdots$$
  1. By considering a suitable geometric series show that \(S < \frac{1}{q}\). [3]
  2. Deduce that \(S \notin \mathbb{Z}\). [2]
You are also given that \(\mathrm{e} = \sum_{r=0}^{\infty} \frac{1}{r!}\).
  1. Assume that \(\mathrm{e} = \frac{p}{q}\), where \(p\) and \(q\) are positive integers. By writing the infinite series for \(\mathrm{e}\) in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that \(\mathrm{e}\) is irrational. [3]
OCR MEI Further Extra Pure Specimen Q1
10 marks Challenging +1.8
The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.
  1. Show that \(G\) is a cyclic group generated by the element 4. [3]
  2. Find another generator for \(G\). Justify your answer. [2]
  3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]
OCR MEI Further Extra Pure Specimen Q2
4 marks Challenging +1.2
A binary operation \(*\) is defined on the set \(S = \{p, q, r, s, t\}\) by the following composition table.
\(*\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(q\)\(q\)\(p\)\(s\)\(t\)\(r\)
\(r\)\(r\)\(t\)\(p\)\(q\)\(s\)
\(s\)\(s\)\(r\)\(t\)\(p\)\(q\)
\(t\)\(t\)\(s\)\(q\)\(r\)\(p\)
Determine whether \((S, *)\) is a group. [4]
OCR MEI Further Extra Pure Specimen Q3
12 marks Challenging +1.2
  1. Find the general solution of $$u_n = 8u_{n-1} - 16u_{n-2}, \quad n \geq 2. \quad (*)$$ [4]
A new sequence \(v_n\) is defined by \(v_n = \frac{u_n}{u_{n-1}}\) for \(n \geq 1\).
  1. (A) Use (*) to show that \(v_n = 8 - \frac{16}{v_{n-1}}\) for \(n \geq 2\). [2] (B) Deduce that if \(v_n\) tends to a limit then it must be 4. [2]
  2. Use your general solution in part (i) to show that \(\lim_{n \to \infty} v_n = 4\). [3]
  3. Deduce the value of \(\lim_{n \to \infty} \left(\frac{u_n}{u_{n-2}}\right)\). [1]
OCR MEI Further Extra Pure Specimen Q4
16 marks Challenging +1.8
A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).
  1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
  2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
  3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
  4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]
The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).
  1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]
OCR MEI Further Extra Pure Specimen Q5
18 marks Challenging +1.8
In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.
  1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
  2. Find a vector equation for the axis of the rotation. [4]
  3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
  4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
  5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]
WJEC Unit 1 2019 June Q01
6 marks Challenging +1.2
Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\tan\theta + 2\cos\theta = 0$$ [6]
WJEC Unit 1 2019 June Q02
7 marks Standard +0.3
Find all the values of \(k\) for which the equation \(x^2 + 2kx + 9k = -4x\) has two distinct real roots. [7]
WJEC Unit 1 2019 June Q03
6 marks Standard +0.3
Use an algebraic method to solve the equation \(12x^3 - 29x^2 + 7x + 6 = 0\). Show all your working. [6]