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OCR S2 Q7

10 marks Standard +0.3

7 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 8 ^ { 2 } \right)$. The mean of a random sample of 12 observations of $X$ is denoted by $\bar { X }$. A test is carried out at the $1 \%$ significance level of the null hypothesis $\mathrm { H } _ { 0 } : \mu = 80$ against the alternative hypothesis $\mathrm { H } _ { 1 } : \mu < 80$. The test is summarised as follows: 'Reject $\mathrm { H } _ { 0 }$ if $\bar { X } < c$; otherwise do not reject $\mathrm { H } _ { 0 } { } ^ { \prime }$.

Calculate the value of $c$.
Assuming that $\mu = 80$, state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
1. $\bar { X } = 74.0$,
2. $\bar { X } = 75.0$.
3. Independent repetitions of the above test, using the value of $c$ found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of $\mu$.

OCR S2 Q8

15 marks Moderate -0.3

8 A continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where $n$ and $k$ are positive constants.

Find $k$ in terms of $n$.
Show that $\mathrm { E } ( X ) = \frac { n + 1 } { n + 2 }$. It is given that $n = 3$.
Find the variance of $X$.
One hundred observations of $X$ are taken, and the mean of the observations is denoted by $\bar { X }$. Write down the approximate distribution of $\bar { X }$, giving the values of any parameters.
Write down the mean and the variance of the random variable $Y$ with probability density function given by $$g ( y ) = \begin{cases} 4 \left( y + \frac { 4 } { 5 } \right) ^ { 3 } & - \frac { 4 } { 5 } \leqslant y \leqslant \frac { 1 } { 5 } \\ 0 & \text { otherwise } \end{cases}$$

OCR FP2 Q1

6 marks Standard +0.3

Write down and simplify the first three non-zero terms of the Maclaurin series for $\ln ( 1 + 3 x )$.
Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.

Use the copy of the sketch on the insert to show how an iteration of the form $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$, with starting value $x _ { 1 }$ such that $0 < x _ { 1 } < \alpha$ as shown, converges to the root $x = \alpha$.
State what happens in the iteration in the following two cases.
1. $x _ { 1 }$ is chosen such that $\alpha < x _ { 1 } < \beta$.
2. $x _ { 1 }$ is chosen such that $x _ { 1 } > \beta$. \section*{Jan 2006} 4
  1. \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
  2. (a) $\_\_\_\_$
  (b) $\_\_\_\_$ \section*{Jan 2006}

OCR FP2 Q5

8 marks Challenging +1.2

Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
Show that $y$ cannot take values between - 3 and 1 .

OCR FP2 Q6

8 marks Standard +0.8

It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for $n \geqslant 1$, $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
Evaluate $I _ { 3 }$, giving the answer in terms of e.

OCR FP2 Q7

9 marks Challenging +1.2

7 \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation $y = \sqrt { x }$. A set of $N$ rectangles of unit width is drawn, starting at $x = 1$ and ending at $x = N + 1$, where $N$ is an integer (see diagram).

By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
Hence find, in terms of $N$, limits between which $\sum _ { r = 1 } ^ { N } \sqrt { r }$ lies. \section*{Jan 2006}

OCR FP2 Q8

13 marks Challenging +1.2

8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

State the greatest value of $r$ and the corresponding values of $\theta$.
Find the equations of the tangents at the pole.
Find the exact area enclosed by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 2 } \pi$.
Find, in simplified form, the cartesian equation of the curve.

OCR FP2 Q9

12 marks Standard +0.3

Using the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, prove that $$\sinh 2 x = 2 \sinh x \cosh x$$
Show that the curve with equation $$y = \cosh 2 x - 6 \sinh x$$ has just one stationary point, and find its $x$-coordinate in logarithmic form. Determine the nature of the stationary point.

OCR D2 2006 June Q1

14 marks Standard +0.3

1 The network represents a system of pipes along which fluid can flow from $S$ to $T$. The values on the arcs are lower and upper capacities in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-02_696_1292_376_424}

Calculate the capacity of the cut with $\mathrm { X } = \{ S , A , B , C \} , \mathrm { Y } = \{ D , E , F , G , H , I , T \}$.
Show that the capacity of the cut $\alpha$, shown on the diagram, is 12 litres per second and calculate the minimum flow across the cut $\alpha$, from $S$ to $T$, (without regard to the remainder of the diagram).
Explain why the arc SC must have at least 5 litres per second flowing through it. By considering the flow through $A$, explain why $A D$ cannot be full to capacity.
Show that it is possible for 11 litres per second to flow through the system.
From your previous answers, what can be deduced about the maximum flow through the system?

OCR D2 2006 June Q2

15 marks Moderate -0.3

2 A delivery company needs to transport heavy loads from its warehouse to a ferry port. Each of the roads that it can use has a bridge with a maximum weight limit. The directed network below represents the roads that can be used to get from the warehouse to the ferry port. Road junctions are labelled with (stage; state) labels. The weights on the arcs represent weight limits in tonnes. \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-03_896_1561_468_292}

Explain what a maximin route is.
Set up a dynamic programming tabulation, working backwards from stage 1, to find the two maximin routes through the network. What is the maximum load that can be transported in one journey from the warehouse to the ferry port?
A new road is opened connecting ( $2 ; 0$ ) and ( $2 ; 1$ ). There is no bridge on this road so it does not restrict the weight of the load that can be carried. Using the new road, what is the maximum load that can be transported in one journey from the warehouse to the ferry port? State the route that the delivery company should use. (Do not attempt to apply dynamic programming in this part.)

OCR D2 2006 June Q3

14 marks Standard +0.3

3 Rose and Colin repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Rose for each combination of strategies.

\multirow{6}{*}{Rose's strategy}	Colin's strategy
		$W$	$X$	$Y$	$Z$
	$A$	-1	4	-3	2
	$B$	5	-2	5	6
	C	3	-4	-1	0
	$D$	-5	6	-4	-2

What is the greatest number of points that Colin can win when Rose plays strategy $A$ and which strategy does Colin need to play to achieve this?
Show that strategy $B$ dominates strategy $C$ and also that strategy $Y$ dominates strategy $Z$. Hence reduce the game to a $3 \times 3$ pay-off matrix.
Find the play-safe strategy for each player on the reduced game. Is the game stable? Rose makes a random choice between the strategies, choosing strategy $A$ with probability $p _ { 1 }$, strategy $B$ with probability $p _ { 2 }$ and strategy $D$ with probability $p _ { 3 }$. She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 5 , \\ \text { subject to } & m \leqslant 4 p _ { 1 } + 10 p _ { 2 } , \\ & m \leqslant 9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 } , \\ & m \leqslant 2 p _ { 1 } + 10 p _ { 2 } + p _ { 3 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 , \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ (You are not required to solve this problem.)
Explain how $9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 }$ was obtained. A computer gives the solution to the LP problem as $p _ { 1 } = \frac { 7 } { 48 } , p _ { 2 } = \frac { 27 } { 48 } , p _ { 3 } = \frac { 14 } { 48 }$.
Calculate the value of $M$ at this solution.

OCR D2 2006 June Q4

14 marks Moderate -0.5

4 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in hours). \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-05_680_1125_424_244} (ii) Key: \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_154_225_1119_1509} \captionsetup{labelformat=empty} \caption{Early event Late event time time}

\end{figure}

\includegraphics[max width=\textwidth, alt={}]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_762_1371_1409_427}

Minimum completion time = $\_\_\_\_$ hours Critical activities: $\_\_\_\_$ (iii) $\_\_\_\_$ (iv) \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-11_513_1189_543_520} Number of workers required = $\_\_\_\_$

	$A \bullet$
$B \bullet$	$\bullet J$
$C \bullet$	$\bullet K$
$D \bullet$	$\bullet L$
$E \bullet$	$\bullet M$
$F \bullet$	$\bullet N$

$\_\_\_\_$

	$J$	$K$	$L$	$M$	$N$	$O$
$A$	2	5	2	2	5	2
$B$	2	5	2	0	5	5
$C$	5	0	5	5	2	2
$D$
$E$
$F$

Answer part (iv) in your answer booklet.

OCR D2 2010 June Q1

6 marks Moderate -0.8

1 The famous fictional detective Agatha Parrot is investigating a murder. She has identified six suspects: Mrs Lemon $( L )$, Prof Mulberry $( M )$, Mr Nutmeg $( N )$, Miss Olive $( O )$, Capt Peach $( P )$ and Rev Quince $( Q )$. The table shows the weapons that could have been used by each suspect.

Suspect
		$L$	M	$N$	$O$	$P$	$Q$
Axe handle	A		✓	✓			✓
Broomstick	$B$		✓		✓
Drainpipe	D	✓		✓
Fence post	$F$			✓	✓
Golf club	$G$	✓				✓	✓
Hammer	$H$	✓			✓	✓

Draw a bipartite graph to represent this information. Put the weapons on the left-hand side and the suspects on the right-hand side. Agatha Parrot is convinced that all six suspects were involved, and that each used a different weapon. She originally thinks that the axe handle was used by Prof Mulberry, the broomstick by Miss Olive, the drainpipe by Mrs Lemon, the fence post by Mr Nutmeg and the golf club by Capt Peach. However, this would leave the hammer for Rev Quince, which is not a possible pairing.
Draw a second bipartite graph to show this incomplete matching.
Construct the shortest possible alternating path from $H$ to $Q$ and hence find a complete matching. Write down which suspect used each weapon.
Find a different complete matching in which none of the suspects used the same weapon as in the matching from part (iii).

OCR D2 2010 June Q2

10 marks Moderate -0.5

2 In an investigation into a burglary, Agatha has five suspects who were all known to have been near the scene of the crime, each at a different time of the day. She collects evidence from witnesses and draws up a table showing the number of witnesses claiming sight of each suspect near the scene of the crime at each possible time. Suspect \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Time}

		1 pm	2 pm	3 pm	4 pm	5 pm
Mrs Rowan	$R$	3	4	2	7	1
Dr Silverbirch	$S$	5	10	6	6	6
Mr Thorn	$T$	4	7	3	5	3
Ms Willow	$W$	6	8	4	8	3
Sgt Yew	$Y$	8	8	7	4	3

\end{table}

Use the Hungarian algorithm on a suitably modified table, reducing rows first, to find the matchings for which the total number of claimed sightings is maximised. Show your working clearly. Write down the resulting matchings between the suspects and the times. Further enquiries show that the burglary took place at 5 pm , and that Dr Silverbirch was not the burglar.
Who should Agatha suspect?

OCR D2 2010 June Q3

11 marks Standard +0.3

Set up a dynamic programming tabulation to find the minimum weight route from ( $0 ; 0$ ) to ( $4 ; 0$ ) on the following directed network. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-03_707_1342_1594_443} Give the route and its total weight.
Explain carefully how the route is obtained directly from the values in the table, without referring to the network.

OCR D2 2010 June Q4

15 marks Moderate -0.3

4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.

	Wai Mai
\cline { 2 - 5 }	$X$	$Y$	$Z$
$A$	2	- 5	3
\cline { 2 - 5 } $E u a n$	- 1	- 3	4
\cline { 1 - 5 } $C$	3	- 5	2
$D$	3	- 2	- 1

Explain what the term 'zero-sum game' means.
How many points does Wai Mai score if she chooses $X$ and Euan chooses $A$ ?
Why should Wai Mai never choose strategy $Z$ ?
Delete the column for $Z$ and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know. The value 3 in the cell corresponding to Euan choosing $D$ and Wai Mai choosing $X$ is changed to - 5 ; otherwise the table is unchanged. Wai Mai decides that she will choose her strategy by making a random choice between $X$ and $Y$, choosing $X$ with probability $p$ and $Y$ with probability $1 - p$.
Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
Using graph paper, draw a graph showing Wai Mai's expected score against $p$ for each of Euan's four strategies and hence calculate the optimum value of $p$.

OCR D2 2010 June Q5

15 marks Standard +0.3

5 Answer this question on the insert provided. The network represents a system of irrigation channels along which water can flow. The weights on the arcs represent the maximum flow in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-05_597_1553_479_296}

Calculate the capacity of the cut that separates $\{ S , B , C , E \}$ from $\{ A , D , F , G , H , T \}$.
Explain why neither arc $S C$ nor arc $B C$ can be full to capacity. Explain why the arcs $E F$ and $E H$ cannot both be full to capacity. Hence find the maximum flow along arc $H T$. When arc $H T$ carries its maximum flow, what is the flow along arc $H G$ ?
Show a flow of 58 litres per second on the diagram in the insert, and find a cut of capacity 58. The direction of flow in $H G$ is reversed.
Use the diagram in the insert to show the excess capacities and potential backflows for your flow from part (iii) in this case.
Without augmenting the labels from part (iv), write down flow augmenting routes to enable an additional 2 litres per second to flow from $S$ to $T$.
Show your augmented flow on the diagram in the insert. Explain how you know that this flow is maximal.

OCR D2 2010 June Q6

15 marks Standard +0.3

6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}

Complete the table in the insert to show the immediate predecessors for each activity.
Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity $K$ changes to $x$ minutes.
Find, in terms of $x$, expressions for the early event time and the late event time for event 9 .
Find the maximum duration of activity $K$ that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
MATHEMATICS} Decision Mathematics 2
INSERT for Questions 5 and 6 (ii) Dummy activity is needed between event 2 and event 3 because $\_\_\_\_$ Dummy activity is needed between event 4 and event 5 because $\_\_\_\_$ (iii)
Event 1 2 3 4 5 6 7 8 9 10
Early event time
Late event time
Minimum project completion time = $\_\_\_\_$ minutes Critical activities: $\_\_\_\_$ \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
RECOGNISING ACHIEVEMENT

OCR D2 Q1

8 marks Easy -1.2

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-1_762_1475_205_239} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A salesman is planning a four-day trip beginning at his home and ending at town $I$. He will spend the first night in town $A , B$ or $C$, the second night in town $D , E$ or $F$ and the third night in town $G$ or $H$. The network in Figure 1 shows the distances, in tens of miles, that he will drive each day according to the route he chooses. Use dynamic programming to find the shortest route the salesman can take and state the distance he will drive in total using this route.

OCR D2 Q2

9 marks Moderate -0.8

2. Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Stage
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$
Alex	19	69	168
Darren	22	64	157
Leroy	20	72	166
Suraj	23	66	171

Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm.

OCR D2 Q3

10 marks Standard +0.3

3. A project consists of 11 activities, some of which are dependent on others having been completed. The following precedence table summarises the relevant information.

Activity	Depends on	Duration (hours)
A	-	5
B	A	4
C	A	2
D	B, C	11
E	C	4
$F$	D	3
G	D	8
$H$	D, E	2
I	$F$	1
J	$F , G , H$	7
$K$	$I , J$	2

Draw an activity network for the project.
Find the critical path and the minimum time in which the project can be completed. Activity $F$ can be carried out more cheaply if it is allocated more time.
Find the maximum time that can be allocated to activity $F$ without increasing the minimum time in which the project can be completed.

OCR D2 Q4

11 marks Standard +0.3

A sheet is provided for use in answering this question.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_725_1303_274_340} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 above shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.

Calculate the values of cuts $C _ { 1 }$ and $C _ { 2 }$.
Find the minimum cut and state its value. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_645_1316_1430_338} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a feasible flow through the same network.
State the values of $x , y$ and $z$.
Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow. State how you know that you have found a maximal flow.

	\(A \bullet\)
\(B \bullet\)	\(\bullet J\)
\(C \bullet\)	\(\bullet K\)
\(D \bullet\)	\(\bullet L\)
\(E \bullet\)	\(\bullet M\)
\(F \bullet\)	\(\bullet N\)

	\(J\)	\(K\)	\(L\)	\(M\)	\(N\)	\(O\)
\(A\)	2	5	2	2	5	2
\(B\)	2	5	2	0	5	5
\(C\)	5	0	5	5	2	2
\(D\)
\(E\)
\(F\)

\multirow{6}{*}{Rose's strategy}	Colin's strategy
		\(W\)	\(X\)	\(Y\)	\(Z\)
	\(A\)	-1	4	-3	2
	\(B\)	5	-2	5	6
	C	3	-4	-1	0
	\(D\)	-5	6	-4	-2

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