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OCR MEI D2 Q1

16 marks Easy -1.2

The switching circuit in Fig. 1.1 shows switches, s for a car's sidelights, h for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights. \includegraphics{figure_1} (Note: s and h are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)

1. Describe in words the conditions under which the foglights will come on. [2]
Fig. 1.2 shows a combinatorial circuit. \includegraphics{figure_2}
1. Write the output in terms of a Boolean expression involving s, h and f. [2]
2. Use a truth table to prove that $s \wedge h \wedge f = \sim (\sim s \vee \sim h) \wedge f$. [3]
A car's first gear can be engaged (g) if either both the road speed is low (r) and the clutch is depressed (d), or if both the road speed is low (r) and the engine speed is the correct multiple of the road speed (m).
1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent r, and single switches to represent each of d, m and g. [2]
2. Draw a combinatorial circuit to represent the Boolean expression $r \wedge (d \vee m) \wedge g$. [4]
3. Use Boolean algebra to prove that $r \wedge (d \vee m) \wedge g = ((r \wedge d) \vee (r \wedge m)) \wedge g$. [2]
4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch. [1]

OCR MEI D2 Q2

16 marks Moderate -0.8

Karl is considering investing in a villa in Greece. It will cost him 56000 euros (€ 56000). His alternative is to invest his money, £35000, in the United Kingdom. He is concerned with what will happen over the next 5 years. He estimates that there is a 60% chance that a house currently worth € 56000 will appreciate to be worth € 75000 in that time, but that there is a 40% chance that it will be worth only € 55000. If he invests in the United Kingdom then there is a 50% chance that there will be 20% growth over the 5 years, and a 50% chance that there will be 10% growth.

Given that £1 is worth € 1.60, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion. [4]

In fact the £/€ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been 20% growth in the UK then there is a 70% chance that the exchange rate will stand at 1.70 euros per pound, and a 30% chance that it will be 1.50. If growth has been 10% then there is a 40% chance that the exchange rate will stand at 1.70 and a 60% chance that it will be 1.50.

Produce a revised decision tree incorporating this information, and give appropriate advice. [3]

A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for x in cash (in thousands of euros) Karl's utility is $x^{0.5}$, and that for y in property (in thousands of euros), Karl's utility is $y^{0.75}$.

Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice? [3]
Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece. [2]
Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece. [4]

OCR MEI D2 Q3

20 marks Standard +0.3

The distance and route matrices shown in Fig. 3.1 are the result of applying Floyd's algorithm to the incomplete network on 4 vertices shown in Fig. 3.2. \includegraphics{figure_3} \includegraphics{figure_4}

Draw the complete network of shortest distances. [2]
Explain how to use the route matrix to find the shortest route from vertex 4 to vertex 1 in the original incomplete network. [2]

A new vertex, vertex 5, is added to the original network. Its distances from vertices to which it is connected are shown in Fig. 3.3. \includegraphics{figure_5}

Draw the extended network and the complete 5-node network of shortest distances. (You are not required to use an algorithm to find the shortest distances.) [3]
Produce the shortest distance matrix and the route matrix for the extended 5-node network. [3]
Apply the nearest neighbour algorithm to your $5 \times 5$ distance matrix, starting at vertex 1. Give the length of the cycle produced, together with the actual cycle in the original 5-node network. [3]
By deleting vertex 1 and its arcs, and by using Prim's algorithm on the reduced distance matrix, produce a lower bound for the solution to the practical travelling salesperson problem in the original 5-node network. Show clearly your use of the matrix form of Prim's algorithm. [4]
In the original 5-node network find a shortest route starting at vertex 1 and using each of the 6 arcs at least once. Give the length of your route. [3]

OCR MEI D2 Q4

20 marks Standard +0.8

Kassi and Theo are discussing how much oil and how much vinegar to use to dress their salad. They agree to use between 5 and 10ml of oil and between 3 and 6ml of vinegar and that the amount of oil should not exceed twice the amount of vinegar. Theo prefers to have more oil than vinegar. He formulates the following problem to maximise the proportion of oil: Maximise $\frac{x}{x + y}$ subject to $0 \leq x \leq 10$, $0 \leq y \leq 6$, $x - 2y \leq 0$.

Explain why this problem is not an LP. [1]
Use the simplex method to solve the following LP. Maximise $x - y$ subject to $0 \leq x \leq 10$, $0 \leq y \leq 6$, $x - 2y \leq 0$. [7]
Kassi prefers to have more vinegar than oil. She formulates the following LP. Maximise $y - x$ subject to $5 \leq x \leq 10$, $3 \leq y \leq 6$, $x - 2y \leq 0$. Draw separate graphs to show the feasible regions for this problem and for the problem in part (ii). [5]
Explain why the formulation in part (ii) produced a solution for Theo's problem, and why it is more difficult to use the simplex method to solve Kassi's problem in part (iii). [2]
Produce an initial tableau for using the two-stage simplex method to solve Kassi's problem. Explain briefly how to proceed. [5]

Edexcel D2 Q1

6 marks Moderate -0.8

This question should be answered on the sheet provided. The table below shows the distances in miles between five villages. Jane lives in village A and is about to take her daughter's friends home to villages B, C, D and E. She will begin and end her journey at A and wishes to travel the minimum distance possible.

	A	B	C	D	E
A	$-$	4	7	8	2
B	4	$-$	1	5	6
C	7	1	$-$	2	7
D	8	5	2	$-$	3
E	2	6	7	3	$-$

Obtain a minimum spanning tree for the network and hence find an upper bound for the length of Jane's journey. [4 marks]
Using a shortcut, improve this upper bound to find an upper bound of less than 15 miles. [2 marks]

Edexcel D2 Q2

8 marks Standard +0.3

The payoff matrix for player A in a two-person zero-sum game with value V is shown below.

	B
	I	II	III
\multirow{3}{*}{A}	I	6	$-4$	$-1$
	II	$-2$	5	3
	III	5	1	$-3$

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player B.

Rewrite the matrix as necessary and state the new value of the game, v, in terms of V. [2 marks]
Define your decision variables. [2 marks]
Write down the objective function in terms of your decision variables. [2 marks]
Write down the constraints. [2 marks]

Edexcel D2 Q3

9 marks Moderate -0.3

This question should be answered on the sheet provided. The table below gives distances, in miles, for a network relating to a travelling salesman problem.

	A	B	C	D	E	F	G
A	$-$	83	57	68	103	91	120
B	83	$-$	78	63	41	82	52
C	57	78	$-$	37	59	63	74
D	68	63	37	$-$	60	52	62
E	103	41	59	60	$-$	48	51
F	91	82	63	52	48	$-$	77
G	120	52	74	62	51	77	$-$

Use the nearest neighbour algorithm, starting at A, to find an upper bound for the length of a tour beginning and ending at A and state the tour. [4 marks]
By deleting A, obtain a lower bound for the length of a tour. [4 marks]
Hence, write down an inequality which must by satisfied by d, the minimum distance travelled in miles. [1 mark]

Edexcel D2 Q4

10 marks Challenging +1.8

This question should be answered on the sheet provided. A rally consisting of four stages is being planned. The first stage will begin at A and the last stage will end at L. Various routes are being considered, with the end of one stage being the start of the next. The organisers want the shortest stage to be as long as possible. The table below shows the length, in miles, of each of the possible stages.

			Finishing point
		C	D	E	F	G	H	I
\multirow{3}{*}{Starting point}	A	1	4.5	13	10	8	11	4
	B					5	10.5
	C					9	6
	D					12	7	15
	E
	F							5
	G							8
	H							10
	I
	J
	K

Finishing point
J	K	L

		2
9	2	3
2	9
		5
		6
		10

Use dynamic programming to find the route which satisfies the wish of the organisers. State the length of the shortest stage on this route. [10 marks]

Edexcel D2 Q5

11 marks Standard +0.3

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.

	Stage
	1	2	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. [11 marks]

Edexcel D2 Q6

13 marks Moderate -0.3

The payoff matrix for player X in a two-person zero-sum game is shown below.

	Y
	$Y_1$	$Y_2$
\multirow{2}{*}{X}	$X_1$	$-2$	4
	$X_2$	6	$-1$

Explain why the game does not have a saddle point. [3 marks]
Find the optimal strategy for
1. player X, [8 marks]
2. player Y.
Find the value of the game. [2 marks]

Edexcel D2 Q7

18 marks Standard +0.3

A transportation problem has costs, in pounds, and supply and demand, in appropriate units, as given in the transportation tableau below.

	D	E	F	Supply
A	13	11	14	20
B	10	9	12	15
C	15	6	8	25
Demand	30	5	25

Find the initial solution given by the north-west corner rule and state why it is degenerate. [3 marks]
Use the stepping-stone method to obtain an optimal solution minimising total cost. State the resulting transportation pattern and its total cost. [15 marks]

OCR D2 Q1

4 marks Moderate -0.8

The payoff matrix for player $A$ in a two-person zero-sum game is shown below. \begin{array}{c|c|c|c|c} & & \multicolumn{3}{c}{B}
& & \text{I} & \text{II} & \text{III}
\hline \multirow{3}{*}{A} & \text{I} & -3 & 4 & 0
& \text{II} & 2 & 2 & 1
& \text{III} & 3 & -2 & -1
\end{array} Find the optimal strategy for each player and the value of the game. [4 marks]

OCR D2 Q2

12 marks Moderate -0.8

Activity	Time	Precedence
A	5
B	20	A
C	3	A
D	7	A
E	4	B
F	15	C
G	6	C
H	17	D
I	10	F, G
J	2	G, H
K	6	E, I
L	9	I, J
M	3	K, L

Fig. 1 Construct an activity network Use appropriate forward and backward scanning to find

the minimum number of days needed to complete the entire project, [3 marks]
the activities which lie on the critical path. [3 marks]

[6 marks]

Represent this information as a digraph. [3 marks]
Find the minimum cut, expressing it in the form $\{$ $\}|\{$ $\}$, and state its value. [2 marks]
Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow. [5 marks]
Explain how you know that this flow is maximal. [1 mark]

[11 marks]

OCR D2 Q6

42 marks Challenging +1.2

The payoff matrix for player $A$ in a two-person zero-sum game is shown below. \begin{array}{c|c|c|c|c} & & \multicolumn{3}{c}{B}
& & \text{I} & \text{II} & \text{III}
\hline \multirow{2}{*}{A} & \text{I} & -2 & 3 & -1
& \text{II} & 4 & -5 & 2
\end{array}

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$. [7 marks]
By solving this linear programming problem, find the optimal strategy for player $B$ and the value of the game. [14 marks]

[21 marks]

Edexcel AEA 2002 June Q1

8 marks Challenging +1.8

Solve the following equation, for $0 \leq x \leq \pi$, giving your answers in terms of $\pi$. $$\sin 5x - \cos 5x = \cos x - \sin x.$$ [8]

Edexcel AEA 2002 June Q2

9 marks Challenging +1.8

In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of $x^2$ is equal to the coefficient of $x^4$ and the coefficient of $x^3$ is positive. Find the value of $p$. [9]

Edexcel AEA 2002 June Q3

11 marks Challenging +1.8

The curve $C$ has parametric equations $$x = 15t - t^3, \quad y = 3 - 2t^2.$$ Find the values of $t$ at the points where the normal to $C$ at $(14, 1)$ cuts $C$ again. [11]

Edexcel AEA 2002 June Q4

14 marks Hard +2.3

Find the coordinates of the stationary points of the curve with equation $$x^3 + y^3 - 3xy = 48$$ and determine their nature. [14]

Edexcel AEA 2002 June Q5

15 marks Hard +2.3

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation $$y = \sin (\cos x).$$ The curve cuts the $x$-axis at the points $A$ and $C$ and the $y$-axis at the point $B$.

Find the coordinates of the points $A$, $B$ and $C$. [3]
Prove that $B$ is a stationary point. [2]

Given that the region $OCB$ is convex,

show that, for $0 \leq x \leq \frac{\pi}{2}$, $$\sin (\cos x) \leq \cos x$$ and $$(1 - \frac{2}{\pi} x) \sin 1 \leq \sin (\cos x)$$ and state in each case the value or values of $x$ for which equality is achieved. [6]
Hence show that $$\frac{\pi}{4} \sin 1 < \int_0^{\frac{\pi}{2}} \sin(\cos x) \, dx < 1.$$ [4]

Edexcel AEA 2002 June Q6

17 marks Hard +2.3

\includegraphics{figure_2} Figure 2 shows a sketch of part of two curves $C_1$ and $C_2$ for $y \geq 0$. The equation of $C_1$ is $y = m_1 - x^{n_1}$ and the equation of $C_2$ is $y = m_2 - x^{n_2}$, where $m_1$, $m_2$, $n_1$ and $n_2$ are positive integers with $m_2 > m_1$. Both $C_1$ and $C_2$ are symmetric about the line $x = 0$ and they both pass through the points $(3, 0)$ and $(-3, 0)$. Given that $n_1 + n_2 = 12$, find

the possible values of $n_1$ and $n_2$, [4]
the exact value of the smallest possible area between $C_1$ and $C_2$, simplifying your answer, [8]
the largest value of $x$ for which the gradients of the two curves can be the same. Leave your answer in surd form. [5]

Edexcel AEA 2002 June Q7

18 marks Hard +2.3

A student was attempting to prove that $x = \frac{1}{2}$ is the only real root of $$x^3 + \frac{1}{4}x - \frac{1}{2} = 0.$$ The attempted solution was as follows. $$x^3 + \frac{1}{4}x = \frac{1}{2}$$ $$\therefore \quad x(x^2 + \frac{1}{4}) = \frac{1}{2}$$ $$\therefore \quad x = \frac{1}{2}$$ or $$x^2 + \frac{1}{4} = \frac{1}{2}$$ i.e. $$x^2 = -\frac{1}{4} \quad \text{no solution}$$ $$\therefore \quad \text{only real root is } x = \frac{1}{2}$$

Explain clearly the error in the above attempt. [2]
Give a correct proof that $x = \frac{1}{2}$ is the only real root of $x^3 + \frac{1}{4}x - \frac{1}{2} = 0$. [3]

The equation $$x^3 + \beta x - \alpha = 0 \quad \text{(I)}$$ where $\alpha$, $\beta$ are real, $\alpha \neq 0$, has a real root at $x = \alpha$.

Find and simplify an expression for $\beta$ in terms of $\alpha$ and prove that $\alpha$ is the only real root provided $|\alpha| < 2$. [6]

An examiner chooses a positive number $\alpha$ so that $\alpha$ is the only real root of equation (I) but the incorrect method used by the student produces 3 distinct real "roots".

Find the range of possible values for $\alpha$. [7]

Edexcel AEA 2004 June Q1

9 marks Challenging +1.8

Solve the equation $\cos x + \sqrt{(1 - \frac{1}{2} \sin 2x)} = 0$, in the interval $0° \leq x < 360°$. [9]

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	A	B	C	D	E
A	\(-\)	4	7	8	2
B	4	\(-\)	1	5	6
C	7	1	\(-\)	2	7
D	8	5	2	\(-\)	3
E	2	6	7	3	\(-\)

	Y
	\(Y_1\)	\(Y_2\)
\multirow{2}{*}{X}	\(X_1\)	\(-2\)	4
	\(X_2\)	6	\(-1\)

	B
	I	II	III
\multirow{3}{*}{A}	I	6	\(-4\)	\(-1\)
	II	\(-2\)	5	3
	III	5	1	\(-3\)

Questions (33218 questions)

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OCR MEI D2 Q1

OCR MEI D2 Q2

OCR MEI D2 Q3

OCR MEI D2 Q4

Edexcel D2 Q1

Edexcel D2 Q2

Edexcel D2 Q3

Edexcel D2 Q4

Edexcel D2 Q5

Edexcel D2 Q6

Edexcel D2 Q7

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Edexcel AEA 2002 June Q1

Edexcel AEA 2002 June Q2

Edexcel AEA 2002 June Q3

Edexcel AEA 2002 June Q4

Edexcel AEA 2002 June Q5

Edexcel AEA 2002 June Q6

Edexcel AEA 2002 June Q7

Edexcel AEA 2004 June Q1