Questions — OCR MEI (4455 questions)

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OCR MEI Further Mechanics Major Specimen Q8
16 marks Standard +0.3
A tractor has a mass of 6000 kg. When developing a power of 5 kW, the tractor is travelling at a steady speed of 2.5 m s\(^{-1}\) across a horizontal field.
  1. Calculate the magnitude of the resistance to the motion of the tractor. [2]
The tractor comes to horizontal ground where the resistance to motion is different. The power developed by the tractor during the next 10 s has an average value of 8 kW. During this time, the tractor accelerates uniformly from 2.5 m s\(^{-1}\) to 3 m s\(^{-1}\).
    1. Show that the work done against the resistance to motion during the 10 s is 71 750 J. [4]
    2. Assuming that the resistance to motion is constant, calculate its value. [3]
The tractor can usually travel up a straight track inclined at an angle \(\alpha\) to the horizontal, where \(\sin\alpha = \frac{1}{20}\), while accelerating uniformly from 3 m s\(^{-1}\) to 3.25 m s\(^{-1}\) over a distance of 100 m against a resistance to motion of constant magnitude of 2000 N. The tractor develops a fault which limits its maximum power to 16kW.
  1. Determine whether the tractor could now perform the same motion up the track. [You should assume that the mass of the tractor and the resistance to motion remain the same.] [7]
OCR MEI Further Mechanics Major Specimen Q9
14 marks Challenging +1.2
\includegraphics{figure_9} Fig. 9 shows the instant of impact of two identical uniform smooth spheres, A and B, each with mass \(m\). Immediately before they collide, the spheres are sliding towards each other on a smooth horizontal table in the directions shown in the diagram, each with speed \(v\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Show that, immediately after the collision, the speed of A is \(\frac{1}{8}v\). Find its direction of motion. [6]
  2. Find the percentage of the original kinetic energy that is lost in the collision. [7]
  3. State where in your answer to part (i) you have used the assumption that the contact between the spheres is smooth. [1]
OCR MEI Further Mechanics Major Specimen Q10
14 marks Standard +0.3
In this question take \(g = 10\). A smooth ball of mass 0.1 kg is projected from a point on smooth horizontal ground with speed 65 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \frac{3}{4}\). While it is in the air the ball is modelled as a particle moving freely under gravity. The ball bounces on the ground repeatedly. The coefficient of restitution for the first bounce is 0.4.
  1. Show that the ball leaves the ground after the first bounce with a horizontal speed of 52 m s\(^{-1}\) and a vertical speed of 15.6 m s\(^{-1}\). Explain your reasoning carefully. [4]
  2. Calculate the magnitude of the impulse exerted on the ball by the ground at the first bounce. [2]
Each subsequent bounce is modelled by assuming that the coefficient of restitution is 0.4 and that the bounce takes no time. The ball is in the air for \(T_1\) seconds between projection and bouncing the first time, \(T_2\) seconds between the first and second bounces, and \(T_n\) seconds between the \((n-1)\)th and \(n\)th bounces.
    1. Show that \(T_1 = \frac{39}{5}\). [2]
    2. Find an expression for \(T_n\) in terms of \(n\). [2]
  2. According to the model, how far does the ball travel horizontally while it is still bouncing? [3]
  3. According to the model, what is the motion of the ball after it has stopped bouncing? [1]
OCR MEI Further Mechanics Major Specimen Q11
16 marks Challenging +1.2
The region bounded by the \(x\)-axis and the curve \(y = \frac{1}{2}k(1-x^2)\) for \(-1 \leq x \leq 1\) is occupied by a uniform lamina, as shown in Fig. 11.1. \includegraphics{figure_11_1}
  1. In this question you must show detailed reasoning. Show that the centre of mass of the lamina is at \(\left(0, \frac{1}{5}k\right)\). [7]
A shop sign is modelled as a uniform lamina in the form of the lamina in part (i) attached to a rectangle ABCD, where AB = 2 and BC = 1. The sign is suspended by two vertical wires attached at A and D, as shown in Fig. 11.2. \includegraphics{figure_11_2}
  1. Show that the centre of mass of the sign is at a distance $$\frac{2k^2 + 10k + 15}{10k + 30}$$ from the midpoint of CD. [4]
The tension in the wire at A is twice the tension in the wire at D.
  1. Find the value of \(k\). [5]
OCR MEI Further Mechanics Major Specimen Q12
15 marks Challenging +1.2
Fig. 12 shows \(x\)- and \(y\)- coordinate axes with origin O and the trajectory of a particle projected from O with speed 28 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal. After \(t\) seconds, the particle has horizontal and vertical displacements \(x\) m and \(y\) m. Air resistance should be neglected. \includegraphics{figure_12}
  1. Show that the equation of the trajectory is given by $$\tan^2\alpha - \frac{160}{x}\tan\alpha + \frac{160y}{x^2} + 1 = 0.$$ (*) [5]
    1. Show that if (*) is treated as an equation with \(\tan\alpha\) as a variable and with \(x\) and \(y\) as constants, then (*) has two distinct real roots for \(\tan\alpha\) when \(y < 40 - \frac{x^2}{160}\). [3]
    2. Show the inequality in part (ii)(A) as a locus on the graph of \(y = 40 - \frac{x^2}{160}\) in the Printed Answer Booklet and label it R. [1]
S is the locus of points \((x, y)\) where (*) has one real root for \(\tan\alpha\). T is the locus of points \((x, y)\) where (*) has no real roots for \(\tan\alpha\).
  1. Indicate S and T on the graph in the Printed Answer Booklet. [2]
  2. State the significance of R, S and T for the possible trajectories of the particle. [3]
A machine can fire a tennis ball from ground level with a maximum speed of 28 m s\(^{-1}\).
  1. State, with a reason, whether a tennis ball fired from the machine can achieve a range of 80 m. [1]
OCR MEI Further Statistics Minor Specimen Q1
4 marks Moderate -0.8
A darts player is trying to hit the bullseye on a dart board. On each throw the probability that she hits it is \(0.05\), independently of any other throw.
  1. Find the probability that she hits the bullseye for the first time on her \(10\)th throw. [2]
  2. Find the probability that she does not hit the bullseye in her first \(10\) throws. [1]
  3. Write down the expected number of throws which it takes her to hit the bullseye for the first time. [1]
OCR MEI Further Statistics Minor Specimen Q2
8 marks Moderate -0.8
The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.
\(x\)\(0\)\(1\)\(2\)\(3\)\(4\)
\(P(X = x)\)\(0.05\)\(0.2\)\(0.5\)\(0.2\)\(0.05\)
    1. Explain why \(\text{E}(X) = 2\). [1]
    2. Find \(\text{Var}(X)\). [3]
  2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find
The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
  1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
  2. State an advantage of using random sampling in this context. [1]
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]