Questions - A-Level Maths

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Find the packing that results using each of these algorithms.
1. The next-fit method
2. The first-fit method
3. The first-fit decreasing method
A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity \(M \mathrm {~kg}\), where \(M\) is an integer.
The item of size 23 kg can be split into two items, of sizes \(x \mathrm {~kg}\) and \(( 23 - x ) \mathrm { kg }\), where \(x\) may be any integer value you choose from 1 to 11. No other item can be split.
Determine the smallest value of \(M\) for which four bins are needed to pack these eight items. Explain your reasoning carefully. 3 The diagram shows a simplified map of the main streets in a small town. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
There are no traffic lights at junctions X and Y .
The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.
Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F. \includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
Complete the copy of the table in the Printed Answer Booklet.
Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
- Write down the total length of the minimum spanning tree.
- List which arcs of the original network correspond to the arcs used in your minimum spanning tree.
Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .

Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c). 4 Li and Mia play a game in which they simultaneously play one of the strategies \(\mathrm { X } , \mathrm { Y }\) and Z . The tables show the points won by each player for each combination of strategies.
A negative entry means that the player loses that number of points.

	Mia
	X	Y	Z
\multirow{3}{*}{Li}	X	5	- 6	0
\cline { 2 - 5 }	Y	- 2	3	4
\cline { 2 - 5 }	Z	- 1	4	8
\cline { 2 - 5 }

	Mia
	X	Y	Z
\multirow{2}{*}{Li}	X	4
\cline { 2 - 5 }	Y	11		5
\cline { 2 - 5 }	Z	10	5	1
\cline { 2 - 5 }

The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.

Complete the table in the Printed Answer Booklet to show the points won by Mia.
Convert the game into a zero-sum game, giving the pay-offs for Li .

Use dominance to reduce the pay-off matrix for the game to a \(2 \times 2\) table. You do not need to explain the dominance. Mia knows that Li will choose his play-safe strategy.

Determine which strategy Mia should choose to maximise her points. 5 A linear programming problem is formulated as below. Maximise \(\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }\) subject to \(2 x + 3 y \geqslant 12\) \(x + y \leqslant 10\) \(5 x + 2 y \leqslant 30\) \(x \geqslant 0 , y \geqslant 0\)

Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
Hence determine the maximum value of the objective. The constraint \(x + y \leqslant 10\) is changed to \(x + y \leqslant k\), the other constraints are unchanged.

Determine, algebraically, the value of \(k\) for which the maximum value of \(P\) is 3 . Do not draw on the graph from part (a) and do not use the spare grid.

Determine, algebraically, the other value of \(k\) for which there is a (non-optimal) vertex of the feasible region at which \(P = 3\).
Do not draw on the graph from part (a) and do not use the spare grid. 6 Sarah is having some work done on her garden.
The table below shows the activities involved, their durations and their immediate predecessors. These durations and immediate predecessors are known to be correct.

Activity	Immediate predecessors	Duration (hours)
A Clear site	-	4
B Mark out new design	A	1
C Buy materials, turf, plants and trees	-	3
D Lay paths	B, C	1
E Build patio	B, C	2
F Plant trees	D	1
G Lay turf	D, E	1
H Finish planting	F, G	1

Use a suitable model to determine the following.
- The minimum time in which the work can be completed

The activities with zero float
[0pt] (ii) State one practical issue that could affect the minimum completion time in part (a)(i). [1]

Sarah needs the work to be completed as quickly as possible. There will be at least one activity happening at all times, but it may not always be possible to do all the activities that are needed at the same time.

Determine the earliest and latest times at which building the patio (activity E) could start. There needs to be a 2-hour break after laying the paths (activity D). During this time other activities that do not depend on activity D can still take place.

Describe how you would adapt your model to incorporate the 2-hour break.

OCR Further Additional Pure AS 2024 June Q1

2 marks Easy -1.2

1 In this question you must show detailed reasoning. The number \(N\) is written as 28 A 3 B in base-12 form. Express \(N\) in decimal (base-10) form.

OCR Further Additional Pure AS 2024 June Q2

6 marks Standard +0.3

2 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\). It is given that \(\mathbf { a } = \left( \begin{array} { c } 2 \\ 4 \\ 3 \lambda \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } \lambda \\ - 4 \\ 6 \end{array} \right)\), where \(\lambda\) is a real parameter.

In the case when \(\lambda = 3\), determine the area of triangle \(O A B\).
Determine the value of \(\lambda\) for which \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\).

OCR Further Additional Pure AS 2024 June Q3

12 marks Standard +0.8

3 The surface \(S\) has equation \(z = f ( x , y )\), where \(f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }\) for all real values of \(x\) and \(y\). You are given that \(S\) has a stationary point at the origin, \(O\), and a second stationary point at the point \(P ( a , b , c )\), where \(\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )\).

Determine the values of \(a , b\) and \(c\).
Throughout this part, take the values of \(a\) and \(b\) to be those found in part (a).
1. Evaluate \(\mathrm { f } _ { x }\) at the points \(\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )\) and \(\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )\).
2. Evaluate \(\mathrm { f } _ { y }\) at the points \(\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )\) and \(\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )\).
3. Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of \(S\), given by
  - \(z = f ( x , b )\), for \(| x - a | \leqslant 0.1\),
\(z = f ( a , y )\), for \(| y - b | \leqslant 0.1\).

OCR Further Additional Pure AS 2024 June Q4

5 marks Standard +0.8

4 The first five terms of the Fibonacci sequence, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), where \(n \geqslant 1\), are \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , F _ { 4 } = 3\) and \(F _ { 5 } = 5\).

Use the recurrence definition of the Fibonacci sequence, \(\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }\), to express \(\mathrm { F } _ { \mathrm { n } + 4 }\) in terms of \(\mathrm { F } _ { \mathrm { n } }\) and \(\mathrm { F } _ { \mathrm { n } - 1 }\).
Hence prove by induction that \(\mathrm { F } _ { \mathrm { n } }\) is a multiple of 3 when \(n\) is a multiple of 4 .

OCR Further Additional Pure AS 2024 June Q5

14 marks Challenging +1.2

5 The set \(S\) consists of all \(2 \times 2\) matrices having determinant 1 or - 1 . For instance, the matrices \(\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\) are elements of \(S\). It is given that \(\times _ { \mathbf { M } }\) is the operation of matrix multiplication.

State the identity element of \(S\) under \(\times _ { \mathbf { M } }\). The group \(G\) is generated by \(\mathbf { P }\), under \(\times _ { \mathbf { M } }\).
Determine the order of \(G\). The group \(H\) is generated by \(\mathbf { Q }\) and \(\mathbf { R }\), also under \(\times _ { \mathbf { M } }\).
1. By finding each element of \(H\), determine the order of \(H\).
2. List all the proper subgroups of \(H\).
State whether each of the following statements is true or false. Give a reason for each of your answers.
- \(G\) is abelian
- \(G\) is cyclic
- \(H\) is abelian
- \(H\) is cyclic

OCR Further Additional Pure AS 2024 June Q6

9 marks

6 For positive integers \(n\), let \(f ( n ) = 1 + 2 ^ { n } + 4 ^ { n }\).

1. Given that \(n\) is a multiple of 3 , but not of 9 , use the division algorithm to write down the two possible forms that \(n\) can take.
2. Show that when \(n\) is a multiple of 3 , but not of 9 , \(f ( n )\) is a multiple of 73 .
Determine the value of \(\mathrm { f } ( n )\), modulo 73 , in the case when \(n\) is a multiple of 9 .

OCR Further Additional Pure AS 2024 June Q7

12 marks Standard +0.3

7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.

Let \(n\) be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR Further Additional Pure AS 2021 November Q1

5 marks Moderate -0.3

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3 \\ 0 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0 \\ 4 \\ 0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) respectively, relative to the origin \(O\).

1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
Use a vector product method to calculate the area of triangle \(A B C\).

OCR Further Additional Pure AS 2021 November Q2

4 marks Moderate -0.8

2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).

1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
Write down the equation of any one contour of \(S\) which does not include the origin.

OCR Further Additional Pure AS 2021 November Q3

6 marks Standard +0.8

3 For positive integers \(n\), the sequence of Fibonacci numbers, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), starts with the terms \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , \ldots\) and is given by the recurrence relation \(\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )\).

Show that \(\mathrm { F } _ { 3 \mathrm { k } + 3 } = 2 \mathrm {~F} _ { 3 \mathrm { k } + 1 } + \mathrm { F } _ { 3 \mathrm { k } }\), where \(k\) is a positive integer.
Prove by induction that \(\mathrm { F } _ { 3 n }\) is even for all positive integers \(n\).

OCR Further Additional Pure AS 2021 November Q4

6 marks Standard +0.3

Let \(a = 1071\) and \(b = 67\).
1. Find the unique integers \(q\) and \(r\) such that \(\mathrm { a } = \mathrm { bq } + \mathrm { r }\), where \(q > 0\) and \(0 \leqslant r < b\).
2. Hence express the answer to (a)(i) in the form of a linear congruence modulo \(b\).
Use the fact that \(358 \times 715 - 239 \times 1071 = 1\) to prove that 715 and 1071 are co-prime.

OCR Further Additional Pure AS 2021 November Q5

11 marks Challenging +1.8

5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).

In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)

OCR Further Additional Pure AS 2021 November Q6

11 marks Challenging +1.8

6 The set \(S\) consists of the following four complex numbers. \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\) For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
2. Verify that ( \(S , \bigcirc\) ) is a group.
3. State the order of each element of \(( S , \bigcirc )\).
Write down the only proper subgroup of ( \(S , \bigcirc\) ).
1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
2. List all possible generators of \(( S , \bigcirc )\).

OCR Further Additional Pure AS 2021 November Q7

10 marks Challenging +1.2

Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
Use the standard test for divisibility by 11 to prove the following statements.
1. \(10 ^ { 33 } + 1\) is divisible by 11
2. \(10 ^ { 33 } + 1\) is divisible by 121

OCR Further Additional Pure AS 2021 November Q8

7 marks Challenging +1.8

8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}

OCR MEI Further Mechanics Major 2024 June Q1

4 marks Moderate -0.8

1 A car A of mass 1200 kg is about to tow another car B of mass 800 kg in a straight line along a horizontal road by means of a tow-rope attached between A and B. The tow-rope is modelled as being light and inextensible. Just before the tow-rope tightens, A is travelling at a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B is at rest. Just after the tow-rope tightens, both cars have a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find the value of \(v\).
Calculate the magnitude of the impulse on A when the tow-rope tightens.

OCR MEI Further Mechanics Major 2024 June Q2

13 marks Standard +0.3

2 One end of a light spring is attached to a fixed point. A mass of 2 kg is attached to the other end of the spring. The spring hangs vertically in equilibrium. The extension of the spring is 0.05 m .

Find the stiffness of the spring.
Find the energy stored in the spring.
Find the dimensions of stiffness of a spring. A particle P of mass \(m\) is performing complete oscillations with amplitude \(a\) on the end of a light spring with stiffness \(k\). The spring hangs vertically and the maximum speed \(v\) of P is given by the formula \(\mathrm { v } = \mathrm { Cm } ^ { \alpha } \mathrm { a } ^ { \beta } \mathrm { k } ^ { \gamma }\),
where \(C\) is a dimensionless constant.
Use dimensional analysis to determine \(\alpha , \beta\), and \(\gamma\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{3} \includegraphics[alt={},max width=\textwidth]{dc8515fb-7104-4d3d-a8ea-ada2b75c70a2-3_577_613_248_244}
\end{figure} A circular hole with centre C and radius \(r \mathrm {~m}\), where \(r < 0.5\), is cut in a uniform circular disc with centre O and radius 0.5 m . The hole touches the rim of the disc at A (see diagram). The centre of mass, G , of the remainder of the disc is on the rim of the hole.
Determine the value of \(r\).

OCR MEI Further Mechanics Major 2024 June Q4

8 marks Standard +0.3

4 \includegraphics[max width=\textwidth, alt={}, center]{dc8515fb-7104-4d3d-a8ea-ada2b75c70a2-3_527_866_1354_242} A uniform rod AB has mass 3 kg and length 4 m . The end A of the rod is in contact with rough horizontal ground. The rod rests in equilibrium on a smooth horizontal peg 1.5 m above the ground, such that the rod is inclined at an angle of \(25 ^ { \circ }\) to the ground (see diagram). The rod is in a vertical plane perpendicular to the peg.

Determine the magnitude of the normal contact force between the peg and the rod.
Determine the range of possible values of the coefficient of friction between the rod and the ground.

OCR MEI Further Mechanics Major 2024 June Q5

7 marks Standard +0.3

5 A car of mass 850 kg is travelling along a straight horizontal road. The power developed by the car is constant and is equal to 18 kW . There is a constant resistance to motion of magnitude 600 N .

Find the greatest steady speed at which the car can travel. Later in the journey, while travelling at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car comes to the bottom of a straight hill which is inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 40 } \right)\) to the horizontal. The power developed by the car remains constant at 18 kW . The magnitude of the resistance force is no longer constant but changes such that the total work done against the resistance force in ascending the hill is 103000 J . The car takes 10 seconds to ascend the hill and at the top of the hill the car is travelling at \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Determine the distance the car travels from the bottom to the top of the hill.

OCR MEI Further Mechanics Major 2024 June Q6