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 Paper 3 2024 June Q4
4 In this question you must show detailed reasoning. Determine the exact value of \(\frac { 1 } { \sqrt { 2 } + 1 } + \frac { 1 } { \sqrt { 3 } + \sqrt { 2 } } + \frac { 1 } { 2 + \sqrt { 3 } }\).
OCR MEI Paper 3 2024 June Q5
5 In this question you must show detailed reasoning. Using the substitution \(\mathrm { u } = \mathrm { x } + 1\), find the value of the positive integer \(c\) such that \(\int _ { \mathrm { c } } ^ { \mathrm { c } + 4 } \frac { \mathrm { x } } { ( \mathrm { x } + 1 ) ^ { 2 } } \mathrm { dx } = \ln 3 - \frac { 1 } { 3 }\).
OCR MEI Paper 3 2024 June Q6
6 In this question you must show detailed reasoning. Solve the equation \(\tan x - 3 \cot x = 2\) for values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI Paper 3 2021 November Q11
11 In this question you must show detailed reasoning. The diagram shows triangle ABC , with \(\mathrm { BC } = 8 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 45 ^ { \circ }\).
The point D on AC is such that \(\mathrm { DC } = 5 \mathrm {~cm}\) and \(\mathrm { BD } = 7 \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-7_684_553_1119_258} Determine the exact length of AB .
OCR MEI Paper 3 Specimen Q5
5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-05_878_1000_685_255} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Find the area of the part of the circle below the \(x\)-axis.
OCR MEI Further Pure Core AS 2020 November Q1
1 In this question you must show detailed reasoning. Find \(\sum _ { r = 2 } ^ { 50 } \left( \frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \right)\), expressing the answer as an exact fraction.
OCR MEI Further Pure Core AS 2021 November Q8
8 In this question you must show detailed reasoning. The equation \(\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0\) has roots \(\alpha , \beta\) and \(\frac { \alpha } { \beta }\). Given that \(\alpha > 0\), find, in any order,
  • the roots of the equation,
  • the value of \(k\).
OCR MEI Further Pure Core AS Specimen Q8
8 In this question you must show detailed reasoning.
  1. Explain why all cubic equations with real coefficients have at least one real root.
  2. Points representing the three roots of the equation \(z ^ { 3 } + 9 z ^ { 2 } + 27 z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices.
OCR MEI D1 Q2
2 Answer this question on the insert provided.
  1. Use Dijkstra's algorithm to find the least weight route from A to G in the network shown in Fig. 2.1. Show the order in which you label vertices, give the route and its weight. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5d8d35b7-e4ba-4bc0-93a1-0cae58093a02-003_458_586_525_758} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
    \end{figure}
  2. Fig. 2.2 shows a partially completed application of Kruskal's algorithm to find a minimum spanning tree for the network. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5d8d35b7-e4ba-4bc0-93a1-0cae58093a02-003_417_524_1309_786} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} Complete the algorithm and give the total weight of your minimum spanning tree.
OCR MEI D1 2005 January Q2
2 Answer this question on the insert provided.
  1. Use Dijkstra's algorithm to find the least weight route from \(A\) to \(G\) in the network shown in Fig. 2.1. Show the order in which you label vertices, give the route and its weight. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-3_458_584_525_760} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
    \end{figure}
  2. Fig. 2.2 shows a partially completed application of Kruskal's algorithm to find a minimum spanning tree for the network. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-3_421_533_1307_779} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} Complete the algorithm and give the total weight of your minimum spanning tree.
OCR MEI D1 2005 January Q4
4 Answer this question on the insert provided. The table shows activities involved in a "perm" in a hair salon, their durations and immediate predecessors. \begin{table}[h]
ActivityDuration (mins)Immediate predecessor(s)
Ashampoo5-
Bprepare perm lotion2-
Cmake coffee for customer3-
Dtrim5A
Eclean sink3A
Fput rollers in15D
Gclean implements3D
Happly perm lotion5B, F
Ileave to set20C,H
Jclean lotion pot and spreaders3H
Kneutralise and rinse10I, E
Ldry10K
Mwash up and clean up15K
Nstyle4G, L
\captionsetup{labelformat=empty} \caption{Table 4}
\end{table}
  1. Complete the activity-on-arc network in the insert to represent the precedences.
  2. Perform a forward pass and a backward pass to find early and late event times. Give the critical activities and the time needed to complete the perm.
  3. Give the total float time for the activity \(G\). Activities \(\mathrm { D } , \mathrm { F } , \mathrm { H } , \mathrm { K }\) and N require a stylist.
    Activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { E } , \mathrm { G } , \mathrm { J }\) and M are done by a trainee.
    Activities \(I\) and \(L\) require no-one in attendance.
    A stylist and a trainee are to give a perm to a customer.
  4. Use the chart in the insert to show a schedule for the activities, assuming that all activities are started as early as possible.
  5. Which activity would be better started at its latest start time?
OCR MEI D1 2005 June Q1
1 Answer this question on the insert provided. The nodes in the unfinished graph in Fig. 1 represent six components, A, B, C, D, E, F and the mains. The components are to be joined by electrical cables to the mains. Each component can be joined directly to the mains, or can be joined via other components. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03df7d9e-63d4-48fb-9cf3-e92003f44788-2_486_879_623_607} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The total number of connections that the electrician has to make is the sum of the orders of the nodes in the finished graph.
  1. Draw arcs representing suitable cables so that the electrician has to make as few connections as possible. Give this number. The electrician has a junction box. This can be represented by an eighth node on the graph.
  2. What is the minimum number of connections which the electrician will have to make if he uses the junction box?
  3. The electrician has to make more connections if he uses his junction box. Why might he choose to use it anyway? The electrician decides not to use the junction box. He measures each of the distances between pairs of nodes, and records them in a complete graph. He then constructs a minimum connector for his graph to find which nodes to connect.
  4. Will this result in the minimum number of connections? Justify your answer.
OCR MEI D1 2005 June Q2
2 Answer this question on the insert provided. A maze is constructed by building east/west and north/south walls so that there is a route from the entrance to the exit. The maze is shown in Fig. 2.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03df7d9e-63d4-48fb-9cf3-e92003f44788-3_495_717_470_671} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure} On entering the maze Janet says "I'm always going to keep a hand in contact with the wall on the right." John says "I'm always going to keep a hand in contact with the wall on the left."
  1. On the insert for this question show Janet's route through the maze. On the insert show John's route.
  2. Will these strategies always find a way through such mazes? Justify your answer. In some mazes the objective is to get to a marked point before exiting. An example is shown in Fig. 2.2, where \(\mathbf { X }\) is the marked point. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{03df7d9e-63d4-48fb-9cf3-e92003f44788-3_497_716_1672_669} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} In the maze shown in Fig. 2.2 Janet's algorithm takes her to \(\mathbf { X }\). John's algorithm does not take him to \(\mathbf { X }\). John modifies his algorithm by saying that he will turn his back on the exit if he arrives there without visiting \(\mathbf { X }\). He will then move onwards, continuing to keep a hand in contact with the wall on the left.
  3. Will this modified algorithm take John to the point \(\mathbf { X }\) in the maze in Fig. 2.2?
  4. Will this modified algorithm take John to any marked point in the maze in Fig. 2.2? Justify your answer.
OCR MEI D1 2005 June Q4
8 marks
4 Answer parts (i) and (ii) on the insert provided. Fig. 4 shows a network of roads giving direct connections between a city, C , and 7 towns labelled P to V. The weights on the arcs are distances in kilometres. The local coastline is also shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03df7d9e-63d4-48fb-9cf3-e92003f44788-5_536_828_573_642} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Use Dijkstra's algorithm on the insert to find the shortest distances from each of the towns to the city, C. List those distances and give the shortest route from P to C and from V to C. [8]
  2. Use Kruskal's algorithm to find a minimum connector for the network. List the order in which you include arcs and give the length of your connector. A bridge is built giving a direct road between P and Q of length 12 km .
  3. What effect does the bridge have on the shortest distances from the towns to the city? (You do not need to use an algorithm to answer this part of the question.)
  4. What effect does the bridge have on the minimum connector for the network? (You do not need to use an algorithm to answer this part of the question.)
  5. Before the bridge was built it was possible to travel from P to C using every road once and only once. With the bridge in place, it is possible to travel from a different town to C using every road once and only once. Give this town and justify your answer.
OCR MEI D1 2006 June Q1
2 marks
1 Answer this question on the insert provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c429bfed-9241-409a-9cd5-9553bf16c9df-2_658_739_466_662} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Apply Dijkstra's algorithm to the copy of Fig. 1 in the insert to find the least weight route from A to D. Give your route and its weight.
  2. Arc DE is now deleted. Write down the weight of the new least weight route from A to D , and explain how your working in part (i) shows that it is the least weight.
    [0pt] [2]
OCR MEI D1 2006 June Q6
6 Answer parts (ii)(A) and (iii)(B) of this question on the insert provided. A particular component of a machine sometimes fails. The probability of failure depends on the age of the component, as shown in Table 6.
Year of lifefirstsecondthirdfourthfifthsixth
Probability of failure during year,
given no earlier failure
0.100.050.020.200.200.30
\section*{Table 6} You are to simulate six years of machine operation to estimate the probability of the component failing during that time. This will involve you using six 2-digit random numbers, one for each year.
  1. Give a rule for using a 2-digit random number to simulate failure of the component in its first year of life. Similarly give rules for simulating failure during each of years 2 to 6 .
  2. (A) Use your rules, together with the random numbers given in the insert, to complete the simulation table in the insert. This simulates 10 repetitions of six years operation of the machine. Start in the first column working down cell-by-cell. In each cell enter a tick if there is no simulated failure and a cross if there is a simulated failure. Stop and move on to the next column if a failure occurs.
    (B) Use your results to estimate the probability of a failure occurring. It is suggested that any component that has not failed during the first three years of its life should automatically be replaced.
  3. (A) Describe how to simulate the operation of this policy.
    (B) Use the table in the insert to simulate 10 repetitions of the application of this policy. Re-use the same random numbers that are given in the insert.
    (C) Use your results to estimate the probability of a failure occurring.
  4. How might the reliability of your estimates in parts (ii) and (iii) be improved?
OCR MEI Further Mechanics B AS 2021 November Q3
3 marks
3 In this question you must show detailed reasoning. [In this question you may use the formula: Volume of cone \(= \frac { 1 } { 3 } \times\) base area × height.]
The region between the line \(\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }\), where \(a > 0\), the \(x\)-axis and the \(y\)-axis is rotated about the \(y\)-axis to form a uniform right circular cone C with base radius \(a\).
  1. Show that the centre of mass of C is \(\frac { 3 } { 4 } a\) from its base. The cone C is fixed on top of a uniform cube, B , of edge length \(2 a\), so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} The centre of mass of the combined shape lies on the boundary of C and B .
    The density of \(B\) is not equal to the density of \(C\).
  2. Determine the exact value of \(\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }\).
    [0pt] [3]
OCR MEI Further Mechanics B AS Specimen Q6
6 In this question you must show detailed reasoning. As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_725_449_539_751} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure}
  1. Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
  2. Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\). Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure}
  3. Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.
OCR MEI Further Statistics A AS Specimen Q3
3 In this question you must show detailed reasoning. A student is investigating what people think about organic food. She wishes to see if there is any difference between the opinions of females and males. She takes a random sample of 100 people and asks each of them if they think that organic food is better for their health than non-organic food. She will use the data to conduct a hypothesis test. The table below shows the opinions of these 100 people.
\cline { 3 - 4 } \multicolumn{2}{c|}{}Sex
\cline { 3 - 4 } \multicolumn{2}{c|}{}FemaleMale
\multirow{2}{*}{
Opinion on
organic food
}		Organic better3518
\cline { 2 - 4 }Not better2225
  1. Explain why the student should use a random sample.
  2. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a person's sex and their opinion on organic food. Show your calculations.
OCR MEI Further Pure Core 2019 June Q4
4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01a574f1-f6f6-40f5-baa5-535c36269731-2_501_670_1329_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
OCR MEI Further Pure Core 2019 June Q8
8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
  1. Evaluate, in exact form, the roots of the equation.
  2. Find \(k\).
OCR MEI Further Pure Core 2019 June Q10
10 In this question you must show detailed reasoning.
  1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
  2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
  3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).
OCR MEI Further Pure Core 2019 June Q15
15 In this question you must show detailed reasoning. Show that \(\int _ { \frac { 3 } { 4 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 4 x + 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { 3 + \sqrt { 5 } } { 2 } \right)\).
OCR MEI Further Pure Core 2023 June Q15
15 In this question you must show detailed reasoning. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 1 + 2 x - x ^ { 2 } } } d x\), giving your answer in terms of \(\pi\).
OCR MEI Further Pure Core 2024 June Q16
16 In this question you must show detailed reasoning. Show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } } \mathrm { dx } = \ln \left( \frac { \mathrm { a } + \mathrm { b } \sqrt { 3 } } { \mathrm { c } } \right)\), where \(a , b\) and \(c\) are integers to be determined.