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OCR Further Discrete 2017 Specimen Q1

13 marks Moderate -0.5

Fiona is a mobile hairdresser. One day she needs to visit five clients, A to E, starting and finishing at her own house at F. She wants to find a suitable route that does not involve her driving too far.

Which standard network problem does Fiona need to solve? [1]

The shortest distances between clients, in km, are given in the matrix below.

	A	B	C	D	E
A	-	12	8	6	4
B	12	-	10	8	10
C	8	10	-	13	10
D	6	8	13	-	10
E	4	10	10	10	-

Use the copy of the matrix in the Printed Answer Booklet to construct a minimum spanning tree for these five client locations. State the algorithm you have used, show the order in which you build your tree and give its total weight. Draw your minimum spanning tree. [4]

The distance from Fiona's house to each client, in km, is given in the table below.

	A	B	C	D	E
F	2	11	9	7	5

Use this information together with your answer to part (ii) to find a lower bound for the length of Fiona's route. [2]
1. Find all the cycles that result from using the nearest neighbour method, starting at F. [3]
2. Use these to find an upper bound for the length of Fiona's route. [2]
Fiona wants to drive less than 35 km. Using the information in your answers to parts (iii) and (iv) explain whether or not a route exists which is less than 35 km in length. [1]

OCR Further Discrete 2017 Specimen Q2

13 marks Standard +0.3

Kirstie has bought a house that she is planning to renovate. She has broken the project into a list of activities and constructed an activity network, using activity on arc. \includegraphics{figure_1}

Construct a cascade chart for the project, showing the float for each non-critical activity. [7]
Calculate the float for remodelling the internal layout stating how much of this is independent float and how much is interfering float. [3]

Kirstie needs to supervise the project. This means that she cannot allow more than three activities to happen on any day.

Describe how Kirstie should organise the activities so that the project is completed in the minimum project completion time and no more than three activities happen on any day. [3]

OCR Further Discrete 2017 Specimen Q3

9 marks Standard +0.8

Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.

How many different ways are there to pair the letters with the envelopes? [1]
Find the number of arrangements with exactly three letters in the correct envelopes. [2]
1. Show that there are two derangements of the three symbols A, B and C. [1]
2. Hence find the number of arrangements with exactly two letters in the correct envelopes. [1]

Let $D_n$ represent the number of derangements of $n$ symbols.

Explain why $D_n = (n-1) \times (D_{n-1} + D_{n-2})$. [2]
Find the number of ways in which all five letters are in the wrong envelopes. [2]

OCR Further Discrete 2017 Specimen Q4

11 marks Standard +0.8

The table shows the pay-off matrix for player $A$ in a two-person zero-sum game between $A$ and $B$.

	Player $B$
	Strategy $X$	Strategy $Y$	Strategy $Z$
Player $A$ Strategy $P$	4	5	$-4$
Player $A$ Strategy $Q$	3	$-1$	2
Player $A$ Strategy $R$	4	0	2

Find the play-safe strategy for player $A$ and the play-safe strategy for player $B$. Use the values of the play-safe strategies to determine whether the game is stable or unstable. [3]
If player $B$ knows that player $A$ will use their play-safe strategy, which strategy should player $B$ use? [1]
Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable. [2]
Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable. [2]
Show that the zero-sum game with the new pay-off value found in part (iv) has a Nash equilibrium and explain what this means for the players. [3]

OCR Further Discrete 2017 Specimen Q5

13 marks Standard +0.3

A garden centre sells tulip bulbs in mixed packs. The cost of each pack and the number of tulips of each colour are given in the table.

	Cost (£)	Red	White	Yellow	Pink
Pack A	50	25	25	25	25
Pack B	48	40	30	30	0
Pack C	53	20	30	40	10

Dirk is designing a floral display in which he will need the number of red tulips to be at most 50 more than the number of white tulips, and the number of white tulips to be less than or equal to twice the number of pink tulips. He has a budget of £240 and wants to find out which packs to buy to maximise the total number of bulbs. Dirk uses the variables $x$, $y$ and $z$ to represent, respectively, how many of pack A, pack B and pack C he buys. He sets up his problem as an initial simplex tableau, which is shown below.

Initial tableau	$P$	$x$	$y$	$z$	$s$	$t$	$u$	RHS
Row 1	1	$-1$	$-1$	$-1$	0	0	0	0
Row 2	0	0	1	$-1$	1	0	0	5
Row 3	0	$-5$	6	2	0	1	0	0
Row 4	0	50	48	53	0	0	1	240

Show how the constraint on the number of red tulips leads to one of the rows of the tableau. [3]

The tableau that results after the first iteration is shown below.

After first iteration	$P$	$x$	$y$	$z$	$s$	$t$	$u$	RHS
Row 5	1	0	$-0.04$	0.06	0	0	0.02	4.8
Row 6	0	0	1	$-1$	1	0	0	5
Row 7	0	0	10.8	7.3	0	1	0.1	24
Row 8	0	1	0.96	1.06	0	0	0.02	4.8

Which cell was used as the pivot? [1]
Explain why row 2 and row 6 are the same. [1]
1. Read off the values of $x$, $y$ and $z$ after the first iteration. [1]
2. Interpret this solution in terms of the original problem. [2]
Identify the variable that has become non-basic. Use the pivot row of the initial tableau to eliminate $x$ algebraically from the equation represented by Row 1 of the initial tableau. [3]

The feasible region can be represented graphically in three dimensions, with the variables $x$, $y$ and $z$ corresponding to the $x$-axis, $y$-axis and $z$-axis respectively. The boundaries of the feasible region are planes. Pairs of these planes intersect in lines and at the vertices of the feasible region these lines intersect.

The planes defined by each of the new basic variables being set equal to 0 intersect at a point. Show how the equations from part (v) are used to find the values $P$ and $x$ at this point. [2]

OCR Further Discrete 2017 Specimen Q6

16 marks Challenging +1.2

A planar graph $G$ is described by the adjacency matrix below. $$\begin{pmatrix} 0 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \end{pmatrix}$$

Draw the graph $G$. [1]
Use Euler's formula to verify that there are four regions. Identify each region by listing the vertices that define it. [3]
Explain why graph $G$ cannot have a Hamiltonian cycle that includes the edge $AB$. Deduce how many Hamiltonian cycles graph $G$ has. [4]

A colouring algorithm is given below. STEP 1: Choose a vertex, colour this vertex using colour 1. STEP 2: If all vertices are coloured, STOP. Otherwise use colour 2 to colour all uncoloured vertices for which there is an edge that joins that vertex to a vertex of colour 1. STEP 3: If all vertices are coloured, STOP. Otherwise use colour 1 to colour all uncoloured vertices for which there is an edge that joins that vertex to a vertex of colour 2. STEP 4: Go back to STEP 2.

Apply this algorithm to graph $G$, starting at $E$. Explain how the colouring shows you that graph $G$ is not bipartite. [2]

By removing just one edge from graph $G$ it is possible to make a bipartite graph.

Identify which edge needs to be removed and write down the two sets of vertices that form the bipartite graph. [2]

Graph $G$ is augmented by the addition of a vertex $X$ joined to each of $A$, $B$, $C$, $D$, $E$ and $F$.

Apply Kuratowski's theorem to a contraction of the augmented graph to explain how you know that the augmented graph has thickness 2. [4]

Solve the recurrence relation $u_{n+2} = 4u_{n+1} - 4u_n$ for $n \geq 0$, given that $u_0 = 1$ and $u_1 = 1$. [4]
Show that each term of the sequence $\{u_n\}$ is an integer. [2]

OCR Further Additional Pure 2017 Specimen Q5

9 marks Challenging +1.2

In this question you must show detailed reasoning. It is given that $I_n = \int_0^\pi \sin^n \theta \, d\theta$ for $n \geq 0$.

Prove that $I_n = \frac{n-1}{n} I_{n-2}$ for $n \geq 2$. [5]
Evaluate $I_1$ and use the reduction formula to determine the exact value of $\int_0^\pi \cos^2 \theta \sin^5 \theta \, d\theta$. [4]

OCR Further Additional Pure 2017 Specimen Q6

10 marks Challenging +1.2

A surface $S$ has equation $z = f(x, y)$, where $f(x, y) = 2x^2 - y^2 + 3xy + 17y$. It is given that $S$ has a single stationary point, $P$.

Determine the coordinates, and the nature, of $P$. [8]
Find the equation of the tangent plane to $S$ at the point $Q(1, 2, 38)$. [2]

OCR Further Additional Pure 2017 Specimen Q7

11 marks Standard +0.3

In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, $t$ years after their introduction, is denoted by $N_t$. The initial number of breeding pairs is given by $N_0$. An initial discrete population model is proposed for $N_t$. Model I: $N_{t+1} = \frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)$

1. For Model I, show that the steady state values of the number of breeding pairs are 0 and 150. [3]
2. Show that $N_{t+1} - N_t < 150 - N_t$ when $N_t$ lies between 0 and 150. [3]
3. Hence determine the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when $N_0 \in (0, 150)$. [2]
An alternative discrete population model is proposed for $N_t$. Model II: $N_{t+1} = \text{INT}\left(\frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)\right)$
Given that $N_0 = 8$, find the value of $N_4$ for each of the two models and give a reason why Model II may be considered more suitable. [3]

OCR Further Additional Pure 2017 Specimen Q8

13 marks Challenging +1.8

The set $X$ consists of all $2 \times 2$ matrices of the form $\begin{pmatrix} x & -y \\ y & x \end{pmatrix}$, where $x$ and $y$ are real numbers which are not both zero.

1. The matrices $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ and $\begin{pmatrix} c & -d \\ d & c \end{pmatrix}$ are both elements of $X$. Show that $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\begin{pmatrix} c & -d \\ d & c \end{pmatrix} = \begin{pmatrix} p & -q \\ q & p \end{pmatrix}$ for some real numbers $p$ and $q$ to be found in terms of $a$, $b$, $c$ and $d$. [2]
2. Prove by contradiction that $p$ and $q$ are not both zero. [5]
Prove that $X$, under matrix multiplication, forms a group $G$. [You may use the result that matrix multiplication is associative.] [4]
Determine a subgroup of $G$ of order 17. [2]

OCR Further Additional Pure 2017 Specimen Q9

14 marks Hard +2.3

1. Prove that $p \equiv \pm 1 \pmod{6}$ for all primes $p > 3$. [2]
2. Hence or otherwise prove that $p^2 - 1 \equiv 0 \pmod{24}$ for all primes $p > 3$. [3]
Given that $p$ is an odd prime, determine the residue of $2^{p^2-1}$ modulo $p$. [4]
Let $p$ and $q$ be distinct primes greater than 3. Prove that $p^{q-1} + q^{p-1} \equiv 1 \pmod{pq}$. [5]

OCR FP1 AS 2017 Specimen Q1

3 marks Moderate -0.3

**In this question you must show detailed reasoning.** The equation $x^2 + 2x + 5 = 0$ has roots $\alpha$ and $\beta$. The equation $x^2 + px + q = 0$ has roots $\alpha^2$ and $\beta^2$. Find the values of $p$ and $q$. [3]

OCR FP1 AS 2017 Specimen Q2

4 marks Moderate -0.8

**In this question you must show detailed reasoning.** Given that $z_1 = 3 + 2i$ and $z_2 = -1 - i$, find the following, giving each in the form $a + bi$.

$z_1^* z_2$ [2]
$\frac{z_1 + 2z_2}{z_2}$ [2]

OCR FP1 AS 2017 Specimen Q3

9 marks Moderate -0.3

You are given two matrices, **A** and **B**, where $$\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \text{ and } \mathbf{B} = \begin{pmatrix} -1 & 2 \\ 2 & -1 \end{pmatrix}.$$ Show that $\mathbf{AB} = m\mathbf{I}$, where $m$ is a constant to be determined. [2]
You are given two matrices, **C** and **D**, where $$\mathbf{C} = \begin{pmatrix} 2 & 1 & 5 \\ 1 & 1 & 3 \\ -1 & 2 & 2 \end{pmatrix} \text{ and } \mathbf{D} = \begin{pmatrix} -4 & 8 & -2 \\ -5 & 9 & -1 \\ 3 & -5 & 1 \end{pmatrix}.$$ Show that $\mathbf{C}^{-1} = k\mathbf{D}$ where $k$ is a constant to be determined. [2]
The matrices **E** and **F** are given by $\mathbf{E} = \begin{pmatrix} k & k^2 \\ 3 & 0 \end{pmatrix}$ and $\mathbf{F} = \begin{pmatrix} 2 \\ k \end{pmatrix}$ where $k$ is a constant. Determine any matrix **F** for which $\mathbf{EF} = \begin{pmatrix} -2k \\ 6 \end{pmatrix}$. [5]

OCR FP1 AS 2017 Specimen Q4

4 marks Moderate -0.3

Draw the region of the Argand diagram for which $|z - 3 - 4i| \leq 5$ and $|z| \leq |z - 2|$. [4]

OCR FP1 AS 2017 Specimen Q5

9 marks Standard +0.3

The matrix **M** is given by $\mathbf{M} = \begin{pmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{pmatrix}$.

The diagram in the Printed Answer Booklet shows the unit square $OABC$. The image of the unit square under the transformation represented by **M** is $OA'B'C'$. Draw and clearly label $OA'B'C'$. [3]
Find the equation of the line of invariant points of this transformation. [3]
1. Find the determinant of **M**. [1]
2. Describe briefly how this value relates to the transformation represented by **M**. [2]

OCR FP1 AS 2017 Specimen Q6

6 marks Moderate -0.3

At the beginning of the year John had a total of £2000 in three different accounts. He has twice as much money in the current account as in the savings account. • The current account has an interest rate of 2.5% per annum. • The savings account has an interest rate of 3.7% per annum. • The supersaver account has an interest rate of 4.9% per annum. John has predicted that he will earn a total interest of £92 by the end of the year.

Model this situation as a matrix equation. [2]
Find the amount that John had in each account at the beginning of the year. [2]
In fact, the interest John will receive is £92 **to the nearest pound**. Explain how this affects the calculations. [2]

OCR FP1 AS 2017 Specimen Q7

9 marks Challenging +1.2

**In this question you must show detailed reasoning.** It is given that $f(z) = z^3 - 13z^2 + 65z - 125$. The points representing the three roots of the equation $f(z) = 0$ are plotted on an Argand diagram. Show that these points lie on the circle $|z| = k$, where $k$ is a real number to be determined. [9]

OCR FP1 AS 2017 Specimen Q8

5 marks Standard +0.8

Prove that $n! > 2^n$ for $n \geq 4$. [5]

OCR FP1 AS 2017 Specimen Q9

11 marks Standard +0.3

Find the value of $k$ such that $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}$ are perpendicular. [2]
Two lines have equations $l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$ and $l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$. Find the point of intersection of $l_1$ and $l_2$. [4]
The vector $\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}$ is perpendicular to the lines $l_1$ and $l_2$. Find the values of $a$ and $b$. [5]

OCR FS1 AS 2017 Specimen Q1

5 marks Moderate -0.3

Two music critics, $P$ and $Q$, give scores to seven concerts as follows.

Concert	1	2	3	4	5	6	7
Score by critic $P$	12	11	6	13	17	16	14
Score by critic $Q$	9	13	8	14	18	16	20

Calculate Spearman's rank correlation coefficient, $r_s$, for these scores. [4]
Without carrying out a hypothesis test, state what your answer tells you about the views of the two critics. [1]

	Player \(B\)
	Strategy \(X\)	Strategy \(Y\)	Strategy \(Z\)
Player \(A\) Strategy \(P\)	4	5	\(-4\)
Player \(A\) Strategy \(Q\)	3	\(-1\)	2
Player \(A\) Strategy \(R\)	4	0	2

Initial tableau	\(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	RHS
Row 1	1	\(-1\)	\(-1\)	\(-1\)	0	0	0	0
Row 2	0	0	1	\(-1\)	1	0	0	5
Row 3	0	\(-5\)	6	2	0	1	0	0
Row 4	0	50	48	53	0	0	1	240

Concert	1	2	3	4	5	6	7
Score by critic \(P\)	12	11	6	13	17	16	14
Score by critic \(Q\)	9	13	8	14	18	16	20

Questions — OCR (4907 questions)

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OCR Further Discrete 2017 Specimen Q1

OCR Further Discrete 2017 Specimen Q2

OCR Further Discrete 2017 Specimen Q3

OCR Further Discrete 2017 Specimen Q4

OCR Further Discrete 2017 Specimen Q5

OCR Further Discrete 2017 Specimen Q6

OCR Further Additional Pure 2017 Specimen Q1

OCR Further Additional Pure 2017 Specimen Q2

OCR Further Additional Pure 2017 Specimen Q3

OCR Further Additional Pure 2017 Specimen Q4

OCR Further Additional Pure 2017 Specimen Q5

OCR Further Additional Pure 2017 Specimen Q6

OCR Further Additional Pure 2017 Specimen Q7

OCR Further Additional Pure 2017 Specimen Q8

OCR Further Additional Pure 2017 Specimen Q9

OCR FP1 AS 2017 Specimen Q1

OCR FP1 AS 2017 Specimen Q2

OCR FP1 AS 2017 Specimen Q3

OCR FP1 AS 2017 Specimen Q4

OCR FP1 AS 2017 Specimen Q5

OCR FP1 AS 2017 Specimen Q6

OCR FP1 AS 2017 Specimen Q7

OCR FP1 AS 2017 Specimen Q8

OCR FP1 AS 2017 Specimen Q9

OCR FS1 AS 2017 Specimen Q1