Questions - A-Level Maths

OCR FM1 AS 2017 December Q7

12 marks Standard +0.8

7 The masses of two particles $A$ and $B$ are $m$ and $2 m$ respectively. They are moving towards each other on a smooth horizontal table. Just before they collide their speeds are $u$ and $2 u$ respectively. After the collision the kinetic energy of $A$ is 8 times the kinetic energy of $B$. Find the coefficient of restitution between $A$ and $B$. \section*{END OF QUESTION PAPER}

OCR FD1 AS 2017 December Q1

4 marks Moderate -0.8

1 \includegraphics[max width=\textwidth, alt={}, center]{a7bca340-6947-42b5-bc35-e6d429d6bed7-2_953_559_347_753}

Trace through the flowchart above using the input $N = 97531$. You only need to record values when they change.
Explain why the process in the flowchart is finite.

OCR FD1 AS 2017 December Q2

6 marks Moderate -0.8

2 Rahul is decorating a room. He needs to decorate at least $30 \mathrm {~m} ^ { 2 }$ of the walls using paint, panelling or wallpaper. The cost ( $\pounds$ ) and the time required (hours) to decorate $1 \mathrm {~m} ^ { 2 }$ of wall are shown in the table.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Cost	Time
Paint	1.12	0.50
Panelling	4.62	0.34
Wallpaper	1.61	0.28

Rahul wants to complete decorating the walls in no more than 8 hours and wants to minimise the cost.
Set up an LP formulation for Rahul's problem, defining your variables. You are not required to solve this LP problem.

OCR FD1 AS 2017 December Q3

8 marks Standard +0.3

3 The activities involved in a project and their durations are represented in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{a7bca340-6947-42b5-bc35-e6d429d6bed7-3_494_700_306_683}

Carry out a forward pass and a backward pass through the network.
Find the float for each activity. A delay means that the duration of activity E increases to $x$.
Find the values of $x$ for which activity E is not a critical activity.

OCR FD1 AS 2017 December Q4

8 marks Moderate -0.3

4 Tom is planning a day out walking. He wants to start from his sister's house ( S ), then visit three places $\mathrm { A } , \mathrm { B }$ and C (once each, in any order) and then finish at his own house (T).

Complete the graph in the Printed Answer Booklet showing all possible arcs that could be used to plan Tom's walk. Tom needs to keep the total distance that he walks to a minimum, so he weights his graph.
(a) Why would finding the shortest path from S to T , on Tom's network, not necessarily solve Tom's problem?
(b) Why would finding a minimum spanning tree, on Tom's network, not necessarily solve Tom's problem? The distance matrix below shows the direct distances, in km , between places.
S A B C T
n S 0 3 2 5 4
nA 3 0 2 2.5 2
nB 2 2 0 3.2 2.5
n 5 2.5 3.2 0 2
\cline { 2 - 6 } T 4 2 2.5 2 0
\cline { 2 - 6 }
\cline { 2 - 6 }
- Use an appropriate algorithm to find a minimum spanning tree for Tom's network.
- Give the total weight of the minimum spanning tree.
- - Find the route that solves Tom's problem.
- Give the minimum distance that Tom must walk.

OCR FD1 AS 2017 December Q5

7 marks Standard +0.3

5 In each round of a card game two players each have four cards. Every card has a coloured number.

Player A's cards are red 1 , blue 2 , red 3 and blue 4.
Player B's cards are red 1 , red 2 , blue 3 and blue 4 .

Each player chooses one of their cards. The players then show their choices simultaneously and deduce how many points they have won or lost as follows:

If the numbers are the same both players score 0 .
If the numbers are different but are the same colour, the player with the lower value card scores the product of the numbers on the cards.
If the numbers are different and are different colours, the player with the higher value card scores the sum of the numbers on the cards.
The game is zero-sum.
1. Complete the pay-off matrix for this game, with player A on rows.
2. Determine the play-safe strategy for each player.
3. Use dominance to show that player A should not choose red 3 . You do not need to identify other rows or columns that are dominated.
4. Determine, with a reason, whether the game is stable or unstable.

OCR FD1 AS 2017 December Q6

6 marks Easy -1.8

6 This list is to be sorted into decreasing order, ending up with 31 in the first position and 4 in the last position.
15
4
12
23
14
16
27
31 Initially bubble sort is used.

Record the list at the end of the first, second and third passes. You do not need to show the individual swaps in each pass. After the fourth pass the list is:
23
15
16
27
31
14
12

OCR FD1 AS 2017 December Q15

Challenging +1.2

15
4
12
23
14

OCR FD1 AS 2017 December Q16

Moderate -0.8

16
27
31 Initially bubble sort is used.

Record the list at the end of the first, second and third passes. You do not need to show the individual swaps in each pass. After the fourth pass the list is:
23
15
16
27
31
14
12
9
4 The sorting is then continued using shuttle sort on this list.
Record the list at the end of each of the first, second and third passes of shuttle sort. You do not need to show the individual swaps in each pass.
How many passes through shuttle sort are needed to complete the sort? Explain your reasoning.
A digraph is represented by the adjacency matrix below. $$\begin{gathered} \\ \\ \\ \\ \\ \\ \text { from } \end{gathered} \begin{gathered} \\ \mathrm { A } \\ \mathrm {~B} \\ \mathrm { C } \end{gathered} \quad \quad \left( \begin{array} { c c c } 1 & \mathrm {~B} & \mathrm { C } \\ 1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)$$ (a) For each vertex, write down
- the indegree,
- the outdegree.
  (b) Explain how the indegrees and outdegrees show that the digraph is semi-Eulerian.
- A graph is represented by the adjacency matrix below.
$$\left. \begin{array} { c } \\ \mathrm { D } \\ \mathrm { E } \\ \mathrm {~F} \end{array} \quad \begin{array} { c c c } \mathrm { D } & \mathrm { E } & \mathrm {~F} \\ 2 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 0 \end{array} \right)$$ (a) Explain how the numerical entries in the matrix show that this can be drawn as an undirected graph.
(b) Explain how the adjacency matrix shows that this graph is semi-Eulerian.
By referring to the vertex degrees, explain why the loop from A to itself is shown as a 1 in the adjacency matrix in part (i) but the loop from $D$ to itself is shown as 2 in the adjacency matrix in part (ii).
List the 15 partitions of the set $\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } \}$. Amy, Bao, Cal and Dana want to travel by taxi from a meeting to the railway station. They book taxis but only two taxis turn up. Each taxi must have a minimum of one passenger and can carry a maximum of four passengers. Dana jumps into one of the taxis.
Find the number of ways that Amy, Bao and Cal can be split between the two taxis. Amy does not want to travel in the same taxi as Bao.
Determine the number of ways that Amy, Bao and Cal can be split between taxis with this additional restriction. Six people want to travel by taxi from a hotel to the railway station using taxis. There must be a minimum of one passenger and a maximum of four passengers in each taxi. The taxis may be regarded as being indistinguishable.
Calculate how many ways there are of splitting the six people between taxis. \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA

OCR Further Additional Pure AS 2017 December Q1

3 marks Moderate -0.5

1 Solve $12 x \equiv 3 ( \bmod 99 )$.

OCR Further Additional Pure AS 2017 December Q2

4 marks Standard +0.3

2

For non-zero vectors $\mathbf { x }$ and $\mathbf { y }$, explain the geometrical significance of the statement $\mathbf { x } \times \mathbf { y } = \mathbf { 0 }$.
The points $P$ and $Q$ have position vectors $\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }$ and $\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ respectively. Find, in the form $( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }$, the equation of line $P Q$.

OCR Further Additional Pure AS 2017 December Q3

9 marks Standard +0.3

3 The surface with equation $z = x ^ { 3 } + y ^ { 3 } - 6 x y$ has two stationary points; one at the origin and the second at the point $A$. Determine the coordinates of $A$.

OCR Further Additional Pure AS 2017 December Q4

11 marks Challenging +1.2

4

The binary operation is defined on $\mathbb { Z }$ by $a$ b $b = a + b - a b$ for all $a , b \in \mathbb { Z }$. Prove that is associative on $\mathbb { Z }$. The operation ∘ is defined on the set $A = \{ 0,2,3,4,5,6 \}$ by $a \circ b = a + b - a b ( \bmod 7 )$ for all $a , b \in A$.
Complete the Cayley table for $\left( A , { } ^ { \circ } \right)$ given in the Printed Answer Booklet.
Prove that $( A , \circ )$ is a group. You may assume that the operation is associative.
List all the subgroups of $( A , \circ )$.

OCR Further Additional Pure AS 2017 December Q5

7 marks Challenging +1.8

5 Given that $n$ is a positive integer greater than 2 , prove that

$\quad 10201 _ { n }$ is a square number.
$\quad 1221 _ { n }$ is a composite number.

OCR Further Additional Pure AS 2017 December Q6

10 marks Challenging +1.2

6 For real constants $a$ and $b$, the sequence $U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots$ is given by $$U _ { 1 } = a \text { and } U _ { n } = \left( U _ { n - 1 } \right) ^ { 2 } - b \text { for } n \geqslant 2 .$$

Determine the behaviour of the sequence in the case where $a = 1$ and $b = 3$.
In the case where $b = 6$, find the values of $a$ for which the sequence is constant.
In the case where $a = - 1$ and $b = 8$, prove that $U _ { n }$ is divisible by 7 for all even values of $n$.

OCR Further Additional Pure AS 2017 December Q7

7 marks Standard +0.8

7 The points $A ( 0,35,120 ) , B ( 28,21,120 )$ and $C ( 96,35 , - 72 )$ lie on the sphere $S$, with centre $O$ and radius 125 . Triangle $A B C$ is denoted by $\triangle$.

Find, in simplest surd form, the area of $\Delta$. The points $A , B$ and $C$ also form a spherical triangle, $T$, on the surface of $S$. Each 'side' of $T$ is the shorter arc of the circle, centre $O$ and radius 125, which passes through two of the given vertices of $T$. In order to find an approximation to the area of the spherical triangle, $T$ is being modelled by $\Delta$.
(a) State the assumption being made in using this model.
(b) Say, giving a reason, whether the model gives an under-estimate or an over-estimate of the area of $T$.

OCR Further Additional Pure AS 2017 December Q8

9 marks Challenging +1.8

8 A surface $S$ has equation $z = 8 y ^ { 3 } - 6 x ^ { 2 } y + 60 x y - 15 x ^ { 2 } + 186 x - 150 y - 100$.

(a) Find any stationary points of the section of $S$ given by $y = - 3$.
(b) Find any stationary points of the section of $S$ given by $x = - 1$.
Show that the surface $S$ has a least one saddle point. \section*{OCR} Oxford Cambridge and RSA

OCR Further Pure Core 1 2018 March Q1

5 marks Moderate -0.3

1 In this question you must show detailed reasoning.
Find the square roots of $24 + 10 \mathrm { i }$, giving your answers in the form $a + b \mathrm { i }$.

OCR Further Pure Core 1 2018 March Q2

10 marks Standard +0.3

2 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)$.

Find the value of $a$ such that $\mathbf { A B } = \mathbf { B A }$.
Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.
A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.

OCR Further Pure Core 1 2018 March Q3

4 marks Moderate -0.3

3 Prove by mathematical induction that, for all integers $n \geqslant 1 , n ^ { 5 } - n$ is divisible by 5 .

OCR Further Pure Core 1 2018 March Q4

7 marks Standard +0.8

4 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }$ and $\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }$ respectively.

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect.
Find the cartesian equation of the plane that contains $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core 1 2018 March Q5

6 marks Challenging +1.2

5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$

OCR Further Pure Core 1 2018 March Q6

6 marks Moderate -0.5

6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point $O$. Initially the mass hangs at rest vertically below $O$. The mass is then pulled to one side with the string taut and released from rest. $\theta$ is the angle, in radians, that the string makes with the vertical through $O$ at time $t$ seconds and $\theta$ may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$

Write down the general solution to this differential equation.
Initially the pendulum is released from rest at an angle of $\theta _ { 0 }$. Find the particular solution to the equation in this case.
State any limitations on the model.

OCR Further Pure Core 1 2018 March Q7

8 marks Challenging +1.2

7

Using the definition of $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$ \section*{(ii) In this question you must show detailed reasoning.} By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3 .$$ Give your answer in the form $\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }$ where $a , b$ and $c$ are integers.

OCR Further Pure Core 1 2018 March Q8

7 marks Standard +0.8

8 You are given that $\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }$ where $a$ and $b$ are positive constants. The first three terms in the Maclaurin expansion of $\mathrm { f } ( x )$ are $1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }$.

Find the value of $a$ and the value of $b$.
Explain if there is any restriction on the value of $x$ in order for the expansion to be valid.

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OCR FM1 AS 2017 December Q7

OCR FD1 AS 2017 December Q1

OCR FD1 AS 2017 December Q2

OCR FD1 AS 2017 December Q3

OCR FD1 AS 2017 December Q4

OCR FD1 AS 2017 December Q5

OCR FD1 AS 2017 December Q6

OCR FD1 AS 2017 December Q15

OCR FD1 AS 2017 December Q16

OCR Further Additional Pure AS 2017 December Q1

OCR Further Additional Pure AS 2017 December Q2

OCR Further Additional Pure AS 2017 December Q3

OCR Further Additional Pure AS 2017 December Q4

OCR Further Additional Pure AS 2017 December Q5

OCR Further Additional Pure AS 2017 December Q6

OCR Further Additional Pure AS 2017 December Q7

OCR Further Additional Pure AS 2017 December Q8

OCR Further Pure Core 1 2018 March Q1

OCR Further Pure Core 1 2018 March Q2

OCR Further Pure Core 1 2018 March Q3

OCR Further Pure Core 1 2018 March Q4

OCR Further Pure Core 1 2018 March Q5

OCR Further Pure Core 1 2018 March Q6

OCR Further Pure Core 1 2018 March Q7

OCR Further Pure Core 1 2018 March Q8

	S	A	B	C	T
n S	0	3	2	5	4
nA	3	0	2	2.5	2
nB	2	2	0	3.2	2.5
n	5	2.5	3.2	0	2
\cline { 2 - 6 } T	4	2	2.5	2	0
\cline { 2 - 6 }
\cline { 2 - 6 }