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Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation $y = 0$ has a root near $x = 1$. Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]
$n$ 0 1 2 3 4 5 6 7
$\mathrm { x } _ { \mathrm { n } }$ 1 0.958509 0.950084 0.948261 0.94786 0.947772 0.947753 0.947748
\captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]
$x$ $y$
0.9477475 $- 7.79967 \mathrm { E } - 07$
0.9477485 $- 2.90821 \mathrm { E } - 06$
$x$ $y$
0.947745 $4.54066 \mathrm { E } - 06$
0.947755 $- 1.67417 \mathrm { E } - 05$
\captionsetup{labelformat=empty} \caption{Fig. 10.2}
\end{table}
Consider the information in Fig. 10.1 and Fig. 10.2.

OCR MEI Paper 2 2020 November Q14

8 marks Challenging +1.2

14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of $y = \sin x \cos 2 x$ and $y = \frac { 1 } { 2 } - \sin 2 x \cos x$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-16_647_898_404_233} \captionsetup{labelformat=empty} \caption{Fig. 14}

\end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.

OCR MEI Paper 3 2022 June Q10

5 marks Standard +0.3

10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve $\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }$ has a stationary point near $x = 3$.

Verify that the $x$-coordinate of this stationary point is between 2.6 and 2.7.
Show that this stationary point is a maximum turning point.

OCR MEI Paper 3 2024 June Q3

4 marks Standard +0.8

3 In this question you must show detailed reasoning. The diagram shows the curve with equation $y = x ^ { 5 }$ and the square $O A B C$ where the points $A , B$ and $C$ have coordinates $( 1,0 ) , ( 1,1 )$ and $( 0,1 )$ respectively. The curve cuts the square into two parts. \includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-04_658_780_1318_230} Show that the relationship between the areas of the two parts of the square is $\frac { \text { Area to left of curve } } { \text { Area below curve } } = 5$.

OCR MEI Paper 3 2024 June Q4

2 marks Moderate -0.8

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

6 marks Standard +0.3

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

5 marks Standard +0.3

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

5 marks Challenging +1.2

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 marks Standard +0.3

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

3 marks Moderate -0.3

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

7 marks Challenging +1.2

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 D1 Q2

Moderate -0.3

2 Answer this question on the insert provided.

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}
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

8 marks Moderate -0.3

2 Answer this question on the insert provided.

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}
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

16 marks Moderate -0.8

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]

	Activity	Duration (mins)	Immediate predecessor(s)
A	shampoo	5	-
B	prepare perm lotion	2	-
C	make coffee for customer	3	-
D	trim	5	A
E	clean sink	3	A
F	put rollers in	15	D
G	clean implements	3	D
H	apply perm lotion	5	B, F
I	leave to set	20	C,H
J	clean lotion pot and spreaders	3	H
K	neutralise and rinse	10	I, E
L	dry	10	K
M	wash up and clean up	15	K
N	style	4	G, L

\captionsetup{labelformat=empty} \caption{Table 4}

\end{table}

Complete the activity-on-arc network in the insert to represent the precedences.
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.
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.
Use the chart in the insert to show a schedule for the activities, assuming that all activities are started as early as possible.
Which activity would be better started at its latest start time?

OCR MEI D1 2005 June Q1

8 marks Easy -1.8

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.

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.
What is the minimum number of connections which the electrician will have to make if he uses the junction box?
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.
Will this result in the minimum number of connections? Justify your answer.

OCR MEI D1 2005 June Q2

8 marks Moderate -0.8

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."

On the insert for this question show Janet's route through the maze. On the insert show John's route.
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.
Will this modified algorithm take John to the point $\mathbf { X }$ in the maze in Fig. 2.2?
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

16 marks Standard +0.3

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}

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]
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 .
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.)
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.)
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

8 marks Moderate -0.3

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}

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.
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

16 marks Moderate -0.5

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.

Probability of failure during year,

given no earlier failure

\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.

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 .
(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.
(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.
How might the reliability of your estimates in parts (ii) and (iii) be improved?

OCR Further Statistics AS 2018 June Q6

5 marks Standard +0.8

6 In this question you must show detailed reasoning. The random variable $T$ has a binomial distribution. It is known that $\mathrm { E } ( T ) = 5.625$ and the standard deviation of $T$ is 1.875 . Find the values of the parameters of the distribution.

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\(x\)	\(y\)
0.9477475	\(- 7.79967 \mathrm { E } - 07\)
0.9477485	\(- 2.90821 \mathrm { E } - 06\)
\(x\)	\(y\)
0.947745	\(4.54066 \mathrm { E } - 06\)
0.947755	\(- 1.67417 \mathrm { E } - 05\)

\(n\)	0	1	2	3	4	5	6	7
\(\mathrm { x } _ { \mathrm { n } }\)	1	0.958509	0.950084	0.948261	0.94786	0.947772	0.947753	0.947748

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OCR MEI Paper 2 2018 June Q12

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OCR Further Statistics AS 2018 June Q6