Questions — OCR MEI (4301 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI Further Pure Core 2020 November Q11
11 In this question you must show detailed reasoning. In Fig. 11, the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }\) and FA are \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-5_807_872_443_239} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
OCR MEI Further Pure Core Specimen Q15
15 In this question you must show detailed reasoning. Show that $$\int _ { 0 } ^ { \frac { 2 } { 3 } } \operatorname { arsinh } 2 x \mathrm {~d} x = \frac { 2 } { 3 } \ln 3 - \frac { 1 } { 3 }$$
OCR MEI Further Mechanics Major 2019 June Q7
7 In this question you must show detailed reasoning. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-04_503_885_1665_244} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows the curve with equation \(y = \frac { 2 } { 3 } \ln x\). The region R , shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0 , y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(y\)-coordinate of the centre of mass of the solid.
OCR MEI Further Mechanics Major 2022 June Q6
6 In this question the box should be modelled as a particle. A box of mass mkg is placed on a rough slope which makes an angle of \(\alpha\) with the horizontal.
  1. Show that the box is on the point of slipping if \(\mu = \tan \alpha\), where \(\mu\) is the coefficient of friction between the box and the slope. A box of mass 5 kg is pulled up a rough slope which makes an angle of \(15 ^ { \circ }\) with the horizontal. The box is subject to a constant frictional force of magnitude 3 N . The speed of the box increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point A on the slope to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point B on the slope with B higher up the slope than A . The distance AB is 10 m .
    \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-05_535_876_957_255} The pulling force has constant magnitude P N and acts at a constant angle of \(25 ^ { \circ }\) above the slope, as shown in the diagram.
  2. Use the work-energy principle to determine the value of P .
OCR MEI Further Mechanics Major 2022 June Q13
13 In this question take \(\boldsymbol { g = \mathbf { 1 0 }\).} A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 m . The other end of the string is attached to a fixed point O . The particle P rests in equilibrium at a point A with the string vertical.
  1. Show that the distance OA is 0.5 m . At time \(\mathrm { t } = 0 , \mathrm { P }\) is projected vertically downwards from A with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Throughout the subsequent motion, \(P\) experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) ).
  2. Given that the downward displacement of P from A at time t seconds is x metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 200 x = 0$$
  3. Verify that \(\mathrm { x } = \frac { 5 } { 56 } \mathrm { e } ^ { - 2 \mathrm { t } } \sin ( 14 \mathrm { t } )\).
  4. Determine whether the string becomes slack during the motion.
OCR MEI Further Statistics Minor 2020 November Q3
3 In this question you must show detailed reasoning. In a survey into pet ownership, one of the questions was 'Do you own either a cat or a dog (or both)?’. A total of 121 people took part in the survey and you should assume that they form a random sample of people in a particular town. The results, classified by the age of the person being surveyed, are shown in Table 3. \begin{table}[h]
\multirow{2}{*}{}Ownership of cat or dog
Does ownDoes not own
\multirow{2}{*}{Age}Over 45 years3829
Under 45 years2331
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table} Carry out a test at the 10\% significance level to investigate whether, for people in this town, there is any association between age and ownership of a cat or dog.
OCR MEI Further Statistics Major 2020 November Q8
10 marks
8 In this question you must show detailed reasoning. On the manufacturer's website, it is claimed that the average daily electricity consumption of a particular model of fridge is 1.25 kWh (kilowatt hours). A researcher at a consumer organisation decides to check this figure. A random sample of 40 fridges is selected. Summary statistics for the electricity consumption \(x \mathrm { kWh }\) of these fridges, measured over a period of 24 hours, are as follows.
\(\Sigma x = 51.92 \quad \Sigma x ^ { 2 } = 70.57\) Carry out a test at the \(5 \%\) significance level to investigate the validity of the claim on the website.
[0pt] [10]
OCR MEI D2 2006 June Q2
2 Answer this question on the insert provided. Fig. 2 shows a network in which the weights on the arcs represent distances. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9716cf3f-afa5-44a4-a8cd-f7511449d06b-2_405_497_1046_776} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Apply Floyd's algorithm on the insert provided to find the complete network of shortest distances.
  2. Show how to use your final matrices to find the shortest route from vertex \(\mathbf { 1 }\) to vertex 3, together with the length of that route.
  3. Use the nearest neighbour algorithm, starting at vertex 1, to find a Hamilton cycle in the complete network of shortest distances. Give the corresponding cycle in the original network, together with its length.
OCR MEI Further Pure Core AS 2019 June Q8
8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.
  1. Show that \(a = - 2\) and \(b = 2\).
  2. Find the other roots of this equation.
OCR MEI Further Pure Core AS 2022 June Q4
4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.
OCR MEI C1 2007 January Q11
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 2009 January Q13
13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q6
6 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
OCR MEI C1 Q4
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\)
    (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 Q4
4 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
OCR MEI C2 2007 January Q13
13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\).
Number of months after start-up \(( x )\)123456
Profit for this month \(( \pounds y )\)5008001200190030004800
The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.
  1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
  3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
  4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
  5. State one way in which this model is unrealistic.
OCR MEI C2 2009 January Q5
5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-3_979_1077_422_536} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} On the insert, draw the graph of
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 2009 January Q12
12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
Years after proposal accepted \(( t )\)12345
Cost \(( \pounds y\) million \()\)250300360440530
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
  1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
  4. According to this model, what was the estimated cost of the project when it was first accepted?
  5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.
OCR MEI C2 2010 January Q12
12 Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005.
Year195519651975198519952005
Population (millions)131161209277372492
Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a 10 ^ { b t }\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.
  1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis.
  2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log _ { 10 } P\) against \(t\). Draw, by eye, a line of best fit on your graph.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate.
OCR MEI C2 2006 June Q12
12 Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10 ^ { b t }\), where \(t\) is the time in years after 2000.
  1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis.
  2. The table gives the data for the population from 2001 to 2005.
    Year20012002200320042005
    \(t\)12345
    \(P\)79008800100001130012800
    Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(t\). Draw a line of best fit for the data.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Predict the population in 2008 according to this model.
OCR MEI C3 Q9
9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f8be5ab-d241-4027-af26-c49da9394adc-4_401_799_488_593} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. On the Insert sketch graphs of
    (A) \(y = 2 \mathrm { f } ( x )\),
    (B) \(y = \mathrm { f } ( - x )\),
    (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
  2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
  3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
  4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
    Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
  5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
  6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation.
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
  7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation.
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
  8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation.
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
  9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}
OCR MEI C3 Q9
9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.
  1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
  2. The data collected are as follows.
    Time \(( t\) years \()\)01234
    Number of bats, \(P\)100170300340360
    Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
    (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
  3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
  4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
  5. Sketch the graph of \(y = 2 \mathrm { f } ( x )\)
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
  6. Sketch the graph of \(y = \mathrm { f } ( 2 x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.}
  7. Plot \(\ln P\) against \(t\).
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
  8. Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph.
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}
OCR MEI C4 Q6
6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.
OCR MEI D1 2006 January Q5
5 Answer this question on the insert provided. Table 5 specifies a road network connecting 7 towns, A, B, \(\ldots\), G. The entries in Table 5 give the distances in miles between towns which are connected directly by roads. \begin{table}[h]
ABCDEFG
A-10---1215
B10-1520--8
C-15-7--11
D-207-20-13
E---20-179
F12---17-13
G1581113913-
\captionsetup{labelformat=empty} \caption{Table 5}
\end{table}
  1. Using the copy of Table 5 in the insert, apply the tabular form of Prim's algorithm to the network, starting at vertex A. Show the order in which you connect the vertices. Draw the resulting tree, give its total length and describe a practical application.
  2. The network in the insert shows the information in Table 5. Apply Dijkstra's algorithm to find the shortest route from A to E. Give your route and its length.
  3. A tunnel is built through a hill between A and B , shortening the distance between A and B to 6 miles. How does this affect your answers to parts (i) and (ii)?
OCR MEI D1 2006 January Q6
6 Answer part (iv) of this question on the insert provided. There are two types of customer who use the shop at a service station. \(70 \%\) buy fuel, the other \(30 \%\) do not. There is only one till in operation.
  1. Give an efficient rule for using one-digit random numbers to simulate the type of customer arriving at the service station. Table 6.1 shows the distribution of time taken at the till by customers who are buying fuel.
    Time taken (mins)11.522.5
    Probability\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 10 }\)
    \section*{Table 6.1}
  2. Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.
    Time taken (mins)11.522.53
    Probability\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 1 } { 7 }\)\(\frac { 1 } { 7 }\)
    \section*{Table 6.2}
  3. Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
  4. Complete the table using the random numbers which are provided.
  5. Calculate the mean total time spent queuing and paying.