Questions — OCR MEI (4456 questions)

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OCR MEI FP3 2011 June Q4
24 marks Challenging +1.8
4
  1. Show that the set \(G = \{ 1,3,4,5,9 \}\), under the binary operation of multiplication modulo 11 , is a group. You may assume associativity.
  2. Explain why any two groups of order 5 must be isomorphic to each other. The set \(H = \left\{ 1 , \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 4 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 6 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 8 } { 5 } \pi \mathrm { j } } \right\}\) is a group under the binary operation of multiplication of complex numbers.
  3. Specify an isomorphism between the groups \(G\) and \(H\). The set \(K\) consists of the 25 ordered pairs \(( x , y )\), where \(x\) and \(y\) are elements of \(G\). The set \(K\) is a group under the binary operation defined by $$\left( x _ { 1 } , y _ { 1 } \right) \left( x _ { 2 } , y _ { 2 } \right) = \left( x _ { 1 } x _ { 2 } , y _ { 1 } y _ { 2 } \right)$$ where the multiplications are carried out modulo 11 ; for example, \(( 3,5 ) ( 4,4 ) = ( 1,9 )\).
  4. Write down the identity element of \(K\), and find the inverse of the element \(( 9,3 )\).
  5. Explain why \(( x , y ) ^ { 5 } = ( 1,1 )\) for every element \(( x , y )\) in \(K\).
  6. Deduce that all the elements of \(K\), except for one, have order 5. State which is the exceptional element.
  7. A subgroup of \(K\) has order 5 and contains the element (9, 3). List the elements of this subgroup.
  8. Determine how many subgroups of \(K\) there are with order 5 .
OCR MEI FP3 2011 June Q5
24 marks Challenging +1.2
5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows. On the first day, the bond is owned by Boyles or Cayleys, with probabilities \(0.4,0.6\) respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities \(0.07,0.8,0.13\) respectively. If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities \(0.15,0.75,0.1\) respectively. If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
  3. Find the probability that the bond is owned by Boyles on the 10th day.
  4. Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
  5. Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
  6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
  7. Find the probability that the bond is eventually owned by Alpha. The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
  8. Find the new probabilities for the ownership of the bond on the first day.
OCR MEI C4 Q1
Easy -2.5
1 Explain why the number 1836.108 for the ratio Rest mass of electron would be suitable for communication with other civilisations whereas neither the rest mass of the proton nor that of the electron would be.
OCR MEI C4 Q2
Standard +0.8
2 A civilisation which works in base 5 sends out the first 6 digits of \(\pi\) as 3.032 32. Convert this to base 10.
OCR MEI C4 Q3
Easy -1.8
3 Complete this table to show the next 3 values of the iteration $$x _ { n + 1 } = k x _ { n } \left( 1 - x _ { n } \right)$$ in the case when \(k = 3.2\) and \(x _ { 0 } = 0.5\). Give your answers to calculator accuracy.
\(n\)\(x _ { n }\)
00.5
10.8
20.512
3
4
5
OCR MEI C4 Q4
Moderate -0.5
4 Justify the statement that the equation in line 83, $$\frac { \phi } { 1 } = \frac { 1 } { \phi - 1 }$$ has the solution \(\phi = \frac { 1 \pm \sqrt { 5 } } { 2 }\).
OCR MEI C4 Q5
Standard +0.8
5 Justify the statement in line 87 that $$\frac { 1 } { \phi } = \frac { \sqrt { 5 } - 1 } { 2 }$$
OCR MEI C4 Q6
4 marks Challenging +1.2
6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
[0pt] [4]
OCR MEI C4 Q8
Standard +0.3
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{070e9904-12b9-4458-b8f2-60c89b31b828-093_1013_1399_488_372} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)
  1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
  2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
  4. Write down a vector equation of the line RS. Calculate the coordinates of S.
OCR MEI C4 2005 June Q2
6 marks Moderate -0.5
2 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 2005 June Q3
4 marks Easy -1.2
3 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of \(\pi\).
OCR MEI C4 2005 June Q4
5 marks Standard +0.3
4 Fig. 4 shows a sketch of the region enclosed by the curve \(\sqrt { 1 + \mathrm { e } ^ { - 2 x } }\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a1123f8-53cd-4b24-bec6-8c3bccc22653-3_517_755_1576_649} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find the volume of the solid generated when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in an exact form.
OCR MEI C4 2005 June Q5
7 marks Moderate -0.3
5 Solve the equation \(2 \cos 2 x = 1 + \cos x\), for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
OCR MEI C4 2005 June Q6
8 marks Standard +0.3
6 A curve has cartesian equation \(y ^ { 2 } - x ^ { 2 } = 4\).
  1. Verify that $$x = t - \frac { 1 } { t } , \quad y = t + \frac { 1 } { t } ,$$ are parametric equations of the curve.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - 1 ) ( t + 1 ) } { t ^ { 2 } + 1 }\). Hence find the coordinates of the stationary points of the curve. Section B (36 marks)
OCR MEI C4 2005 June Q7
18 marks Standard +0.3
7 In a chemical process, the mass \(M\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = \frac { M } { t \left( 1 + t ^ { 2 } \right) }$$
  1. Find \(\int \frac { t } { 1 + t ^ { 2 } } \mathrm {~d} t\).
  2. Find constants \(A , B\) and \(C\) such that $$\frac { 1 } { t \left( 1 + t ^ { 2 } \right) } = \frac { A } { t } + \frac { B t + C } { 1 + t ^ { 2 } } .$$
  3. Use integration, together with your results in parts (i) and (ii), to show that $$M = \frac { K t } { \sqrt { 1 + t ^ { 2 } } } ,$$ where \(K\) is a constant.
  4. When \(t = 1 , M = 25\). Calculate \(K\). What is the mass of the chemical in the long term?
OCR MEI S1 Q3
8 marks Standard +0.3
3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{1ad9c390-b42f-47d8-86c5-f73a42d97721-02_513_826_1713_658} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).
OCR MEI S1 Q7
18 marks Easy -1.3
7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Distance from school} \includegraphics[alt={},max width=\textwidth]{1ad9c390-b42f-47d8-86c5-f73a42d97721-04_1073_1571_580_340}
\end{figure}
  1. Use the graph to estimate, to the nearest 10 metres,
    (A) the median distance from school,
    (B) the lower quartile, upper quartile and interquartile range.
  2. Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.
    Distance (metres)20040060080010001200
    Cumulative frequency2064118150169176
  3. Copy and complete the grouped frequency table below describing the same data.
    Distance ( \(d\) metres)Frequency
    \(0 < d \leqslant 200\)20
    \(200 < d \leqslant 400\)
  4. Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
  5. Calculate the revised estimate of the mean distance.
  6. Describe what change needs to be made to the cumulative frequency graph.
OCR MEI S2 Q3
18 marks Standard +0.3
3 In a triathlon, competitors have to swim 600 metres, cycle 40 kilometres and run 10 kilometres. To improve her strength, a triathlete undertakes a training programme in which she carries weights in a rucksack whilst running. She runs a specific course and notes the total time taken for each run. Her coach is investigating the relationship between time taken and weight carried. The times taken with eight different weights are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\) represent weight carried in kilograms and time taken in minutes respectively. \includegraphics[max width=\textwidth, alt={}, center]{d138173d-c70c-46db-b9b9-d5f19334c5f1-04_627_1536_630_281} Summary statistics: \(n = 8 , \Sigma x = 36 , \Sigma y = 214.8 , \Sigma x ^ { 2 } = 204 , \Sigma y ^ { 2 } = 5775.28 , \Sigma x y = 983.6\).
  1. Calculate the equation of the regression line of \(y\) on \(x\). On one of the eight runs, the triathlete was carrying 4 kilograms and took 27.5 minutes. On this run she was delayed when she tripped and fell over.
  2. Calculate the value of the residual for this weight.
  3. The coach decides to recalculate the equation of the regression line without the data for this run. Would it be preferable to use this recalculated equation or the equation found in part (i) to estimate the delay when the triathlete tripped and fell over? Explain your answer. The triathlete's coach claims that there is positive correlation between cycling and swimming times in triathlons. The product moment correlation coefficient of the times of twenty randomly selected competitors in these two sections is 0.209 .
  4. Carry out a hypothesis test at the \(5 \%\) level to examine the coach's claim, explaining your conclusions clearly.
  5. What distributional assumption is necessary for this test to be valid? How can you use a scatter diagram to decide whether this assumption is likely to be true?
OCR MEI S3 2008 January Q1
18 marks Moderate -0.3
1
  1. The time (in milliseconds) taken by my computer to perform a particular task is modelled by the random variable \(T\). The probability that it takes more than \(t\) milliseconds to perform this task is given by the expression \(\mathrm { P } ( T > t ) = \frac { k } { t ^ { 2 } }\) for \(t \geqslant 1\), where \(k\) is a constant.
    1. Write down the cumulative distribution function of \(T\) and hence show that \(k = 1\).
    2. Find the probability density function of \(T\).
    3. Find the mean time for the task.
  2. For a different task, the times (in milliseconds) taken by my computer on 10 randomly chosen occasions were as follows. $$\begin{array} { c c c c c c c c c c } 6.4 & 5.9 & 5.0 & 6.2 & 6.8 & 6.0 & 5.2 & 6.5 & 5.7 & 5.3 \end{array}$$ From past experience it is thought that the median time for this task is 5.4 milliseconds. Carry out a test at the \(5 \%\) level of significance to investigate this, stating your hypotheses carefully.
OCR MEI S3 2008 January Q2
18 marks Standard +0.3
2 In the vegetable section of a local supermarket, leeks are on sale either loose (and unprepared) or prepared in packs of 4 . The weights of unprepared leeks are modelled by the random variable \(X\) which has the Normal distribution with mean 260 grams and standard deviation 24 grams. The prepared leeks have had \(40 \%\) of their weight removed, so that their weights, \(Y\), are modelled by \(Y = 0.6 X\).
  1. Find the probability that a randomly chosen unprepared leek weighs less than 300 grams.
  2. Find the probability that a randomly chosen prepared leek weighs more than 175 grams.
  3. Find the probability that the total weight of 4 randomly chosen prepared leeks in a pack is less than 600 grams.
  4. What total weight of prepared leeks in a randomly chosen pack of 4 is exceeded with probability 0.975 ?
  5. Sandie is making soup. She uses 3 unprepared leeks and 2 onions. The weights of onions are modelled by the Normal distribution with mean 150 grams and standard deviation 18 grams. Find the probability that the total weight of her ingredients is more than 1000 grams.
  6. A large consignment of unprepared leeks is delivered to the supermarket. A random sample of 100 of them is taken. Their weights have sample mean 252.4 grams and sample standard deviation 24.6 grams. Find a \(99 \%\) confidence interval for the true mean weight of the leeks in this consignment.
OCR MEI S3 2008 January Q3
18 marks Standard +0.3
3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean \(380 ^ { \circ } \mathrm { C }\). The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.
374.0378.1363.0357.0377.9388.4
379.6372.4362.4377.3385.2370.6
  1. Give three reasons why a \(t\) test would be appropriate.
  2. Carry out the test using a \(5 \%\) significance level. State your hypotheses and conclusion carefully.
  3. Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
  4. Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.
OCR MEI S3 2008 January Q4
18 marks Standard +0.3
4
  1. In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution \(\mathrm { B } ( 5 , p )\) should model these data.
    Number of girlsNumber of families
    032
    1110
    2154
    3125
    463
    516
    1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of \(p\).
    2. Investigate at a \(5 \%\) significance level whether the binomial model with \(p\) estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
  2. A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.
OCR MEI M1 Q2
Standard +0.3
2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-002_478_397_1211_872} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. State what information in the question tells you that
    (A) the tension is the same throughout the string,
    (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
  2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
  3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).
OCR MEI M1 Q5
Moderate -0.3
5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-003_424_472_1599_774} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Draw a labelled diagram showing all the forces acting on the box.
  2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
  3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical. 7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-004_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
  1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
  2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
  3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)
OCR MEI M1 Q7
Standard +0.3
7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-004_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
  2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
  3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)