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AQA D2 2013 June Q3

12 marks Moderate -0.3

3 The table shows the times taken, in minutes, by five people, $A , B , C , D$ and $E$, to carry out the tasks $V , W , X , Y$ and $Z$.

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$
Task $\boldsymbol { V }$	100	110	112	102	95
Task $\boldsymbol { W }$	125	130	110	120	115
Task $\boldsymbol { X }$	105	110	101	108	120
Task $\boldsymbol { Y }$	115	115	120	135	110
Task $\boldsymbol { Z }$	100	98	99	100	102

Each of the five tasks is to be given to a different one of the five people so that the total time for the five tasks is minimised. The Hungarian algorithm is to be used.

By reducing the columns first, and then the rows, show that the new table of values is
0 12 13 2 0
14 21 0 $k$ 9
3 10 0 6 23
0 2 6 20 0
0 0 0 0 7
and state the value of the constant $k$.
Show that the zeros in the table in part (a) can be covered with four lines. Use augmentation twice to produce a table where five lines are required to cover the zeros.
Hence find the possible ways of allocating the five tasks to the five people to achieve the minimum total time.
Find the minimum total time.

AQA D2 2013 June Q4

9 marks Standard +0.3

4 A haulage company, based in town $A$, is to deliver a tall statue to town $K$. The statue is being delivered on the back of a lorry. The network below shows a system of roads. The number on each edge represents the height, in feet, of the lowest bridge on that road. The company wants to ensure that the height of the lowest bridge along the route from $A$ to $K$ is maximised. \includegraphics[max width=\textwidth, alt={}, center]{5123be51-168e-4487-8cd3-33aee9e3b23f-10_869_1593_715_221} Working backwards from $\boldsymbol { K }$, use dynamic programming to find the optimal route when driving from $A$ to $K$. You must complete the table opposite as your solution.

Stage	State	From	Value
1	H	$K$
	I	$K$
	J	K
2

Optimal route is

AQA D2 2013 June Q5

15 marks Easy -2.5

5 Romeo and Juliet play a zero-sum game. The game is represented by the following pay-off matrix for Romeo.

	Juliet
\cline { 2 - 5 }	Strategy	D	E	F
A	4	- 4	0
\cline { 2 - 5 } Romeo	B	- 2	- 5	3
\cline { 2 - 5 }	C	2	1	- 2
\cline { 2 - 5 }
\cline { 2 - 5 }

Find the play-safe strategy for each player.
Show that there is no stable solution.
Explain why Juliet should never play strategy D.
1. Explain why the following is a suitable pay-off matrix for Juliet.
  4 5 - 1
  0 - 3 2
2. Hence find the optimal strategy for Juliet.
3. Find the value of the game for Juliet.

AQA D2 2013 June Q6

11 marks Standard +0.3

Display the following linear programming problem in a Simplex tableau.
Maximise $\quad P = 4 x + 3 y + z$ subject to $$\begin{aligned} & 2 x + y + z \leqslant 25 \\ & x + 2 y + z \leqslant 40 \\ & x + y + 2 z \leqslant 30 \end{aligned}$$ and $x \geqslant 0 , \quad y \geqslant 0 , \quad z \geqslant 0$.
The first pivot to be chosen is from the $x$-column. Perform one iteration of the Simplex method.
1. Perform one further iteration.
2. Interpret your final tableau and state the values of your slack variables.

AQA D2 2013 June Q7

11 marks Moderate -0.5

7 Figure 2 shows a network of pipes. Water from two reservoirs, $R _ { 1 }$ and $R _ { 2 }$, is used to supply three towns, $T _ { 1 } , T _ { 2 }$ and $T _ { 3 }$.
In Figure 2, the capacity of each pipe is given by the number not circled on each edge. The numbers in circles represent an initial flow.

Add a supersource, supersink and appropriate weighted edges to Figure 2. (2 marks)
1. Use the initial flow and the labelling procedure on Figure 3 to find the maximum flow through the network. You should indicate any flow augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
2. State the value of the maximum flow and, on Figure 4, illustrate a possible flow along each edge corresponding to this maximum flow.
Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-18_1077_1246_1475_395}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_1049_1264_308_386}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_835_1011_1738_171}
\end{figure}
\includegraphics[max width=\textwidth, alt={}]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_688_524_1448_1302}

\includegraphics[max width=\textwidth, alt={}]{5123be51-168e-4487-8cd3-33aee9e3b23f-20_2256_1707_221_153}

Find the mean value of f
5
The function g is defined by $$\mathrm { g } ( x ) = \mathrm { f } ( x ) + c \quad 1 \leq x \leq 5$$ The mean value of $g$ is 40
Calculate the value of the constant $c$

AQA Further AS Paper 1 2023 June Q6

6 marks Moderate -0.3

Find and simplify the first five terms in the Maclaurin series for $\mathrm { e } ^ { 2 x }$ 6
Hence, or otherwise, write down the first five terms in the Maclaurin series for $\mathrm { e } ^ { - 2 x }$ 6
Hence, or otherwise, show that the Maclaurin series for $\cosh ( 2 x )$ is $$a + b x ^ { 2 } + c x ^ { 4 } + \ldots$$ where $a$, $b$ and $c$ are rational numbers to be determined.

AQA Further AS Paper 1 2023 June Q7

7 marks Standard +0.3

Show that, for all integers $r$, $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) }$$ 7
Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } = \frac { a n } { b n + c }$$ where $a$, $b$ and $c$ are integers to be determined.
7
Hence, or otherwise, evaluate $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { 99 \times 101 }$$

AQA Further AS Paper 1 2023 June Q8

4 marks Moderate -0.3

8 Abdoallah wants to write the complex number $- 1 + \mathrm { i } \sqrt { 3 }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$ where $r \geq 0$ and $- \pi < \theta \leq \pi$ Here is his method: $$\begin{array} { r l r l } r & = \sqrt { ( - 1 ) ^ { 2 } + ( \sqrt { 3 } ) ^ { 2 } } & & \tan \theta = \frac { \sqrt { 3 } } { - 1 } \\ & = \sqrt { 1 + 3 } & & \Rightarrow \\ & = \sqrt { 4 } & & \tan \theta = - \sqrt { 3 } \\ & = 2 & & \theta = \tan ^ { - 1 } ( - \sqrt { 3 } ) \\ & & \theta = - \frac { \pi } { 3 } \\ & - 1 + i \sqrt { 3 } = 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) \end{array}$$ There is an error in Abdoallah's method. 8

Show that Abdoallah's answer is wrong by writing $$2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right)$$ in the form $x + \mathrm { i } y$ Simplify your answer.
8
Explain the error in Abdoallah's method.
8
Express $- 1 + \mathrm { i } \sqrt { 3 }$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$ 8
Write down the complex conjugate of $- 1 + i \sqrt { 3 }$

AQA Further AS Paper 1 2023 June Q9

11 marks Standard +0.3

9 The matrix $\mathbf { M }$ represents the transformation T and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 p + 1 & 12 \\ p + 2 & p ^ { 2 } - 3 \end{array} \right]$$ 9

In the case when $p = 0$ show that the image of the point $( 4,5 )$ under T is the point $( 64 , - 7 )$ 9
In the case when $p = - 2$ find the gradient of the line of invariant points under $T$ 9
Show that $p = 3$ is the only real value of $p$ for which $\mathbf { M }$ is singular.
The curve $C$ has equation $$y = \frac { 3 x ^ { 2 } + m x + p } { x ^ { 2 } + p x + m }$$ where $m$ and $p$ are integers.
The vertical asymptotes of $C$ are $x = - 4$ and $x = - 1$ The curve $C$ is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-12_867_1102_733_463}

AQA Further AS Paper 1 2023 June Q10

9 marks Standard +0.3

Write down the equation of the horizontal asymptote of $C$ 10
Find the value of $m$ and the value of $p$
10
10
Hence, or otherwise, write down the coordinates of the $y$-intercept of $C$
Without using calculus, show that the line $y = - 1$ does not intersect $C$

AQA Further AS Paper 1 2023 June Q11

8 marks Moderate -0.5

11 A point has Cartesian coordinates $( x , y )$ and polar coordinates $( r , \theta )$ where $r \geq 0$ and $- \pi < \theta \leq \pi$ 11

Express $r$ in terms of $x$ and $y$ 11
Express $x$ in terms of $r$ and $\theta$ 11
The curve $C _ { 1 }$ has the polar equation $$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$ 11 (c) (i) Show that the Cartesian equation of $C _ { 1 }$ can be written as $$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$ where $a$ and $b$ are integers to be determined.
11 (c) (ii) The curve $C _ { 2 }$ has the Cartesian equation $$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$ where $a$ and $b$ take the same values as in part (c)(i). Describe fully a single transformation that maps the curve $C _ { 1 }$ onto the curve $C _ { 2 }$

AQA Further AS Paper 1 2023 June Q12

13 marks Standard +0.3

Show that $( 1 + i ) ^ { 4 } = - 4$ 12
The function f is defined by $$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$ 12 (b) (i) Show that (1+i) is a root of $\mathrm { f } ( \mathrm { z } ) = 0$ 12 (b) (ii) Hence write down another root of $\mathrm { f } ( \mathrm { z } ) = 0$ 12 (b) (iii) One of the linear factors of $\mathrm { f } ( \mathrm { z } )$ is $$( z - ( 1 + i ) )$$ Write down another linear factor and hence, or otherwise, find a quadratic factor of $\mathrm { f } ( \mathrm { z } )$ with real coefficients.
12 (b) (iv) Find another quadratic factor of $\mathrm { f } ( \mathrm { z } )$ with real coefficients.
12 (b) (v) Hence explain why the graph of $y = \mathrm { f } ( x )$ does not intersect the $x$-axis.

AQA Further AS Paper 1 2023 June Q13

10 marks Standard +0.3

Prove by induction that, for all integers $n \geq 1$, $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ [4 marks]
13
Hence, or otherwise, write down a factorised expression for the sum of the first $2 n$ squares $$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
Use the formula in part (a) to write down a factorised expression for the sum of the first $n$ even squares $$2 ^ { 2 } + 4 ^ { 2 } + 6 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
Hence, or otherwise, show that the sum of the first $n$ odd squares is $$a n ( b n - 1 ) ( b n + 1 )$$ where $a$ and $b$ are rational numbers to be determined.

AQA Further AS Paper 1 2023 June Q14

4 marks Standard +0.8

14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers $a$ and $b$ \includegraphics[max width=\textwidth, alt={}]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-21_2491_1755_173_123} number Additional page, if required. Write the question numbers in the left-hand margin. $\_\_\_\_$ number \section*{Additional page, if required. Write the question numbers in the left-hand margin.
Additional page, if required. uestion numbers in the left-hand margin.}

Question number	Additional page, if required. Write the question numbers in the left-hand margin.
	$\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

AQA Further AS Paper 2 Statistics 2021 June Q1

1 marks Easy -1.2

1 The discrete random variable $X$ has $\operatorname { Var } ( X ) = 6.5$ Find $\operatorname { Var } ( 4 X - 2 )$ Circle your answer.
2426102104

AQA Further AS Paper 2 Statistics 2021 June Q2

1 marks Easy -1.2

2 The random variable $A$ has a Poisson distribution with mean 2 The random variable $B$ has a Poisson distribution with standard deviation 4 The random variables $A$ and $B$ are independent.
State the distribution of $A + B$ Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)

AQA Further AS Paper 2 Statistics 2021 June Q3

5 marks Moderate -0.8

3 The random variable $X$ has a discrete uniform distribution and takes values $1,2,3 , \ldots , n$ The mean of $X$ is 8 3

Show that $n = 15$ [0pt] [2 marks]
LL
3
$\quad$ Find $\mathrm { P } ( X > 4 )$ 3
Find the variance of $X$, giving your answer in exact form.

AQA Further AS Paper 2 Statistics 2021 June Q4

7 marks Standard +0.3

4 The distance a particular football player runs in a match is modelled by a normal distribution with standard deviation 0.3 kilometres. A random sample of $n$ matches is taken.
The distance the player runs in this sample of matches has mean 10.8 kilometres.
The sample is used to construct a $93 \%$ confidence interval for the mean, of width 0.0543 kilometres, correct to four decimal places. 4

Find the value of $n$ 4
Find the $93 \%$ confidence interval for the mean, giving the limits to three decimal places.
4
Alison claims that the population mean distance the player runs is 10.7 kilometres. She carries out a hypothesis test at the 7\% level of significance using the random sample and the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 10.7 \\ & \mathrm { H } _ { 1 } : \mu \neq 10.7 \end{aligned}$$ 4 (c) (i) State, with a reason, whether the null hypothesis will be accepted or rejected. 4 (c) (ii) Describe, in the context of the hypothesis test in part (c)(i), what is meant by a Type II error. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-06_2488_1730_219_141}

AQA Further AS Paper 2 Statistics 2021 June Q5

5 marks Easy -1.2

5 In a game it is known that:

25\% of players score 0
30\% of players score 5
35\% of players score 10
10\% of players score 20

Players receive prize money, in pounds, equal to 100 times their score.
5

State the modal score.
[0pt] [1 mark] 5
Find the median score.
5
Find the mean prize money received by a player.

AQA Further AS Paper 2 Statistics 2021 June Q6

11 marks Standard +0.3

6 The continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 114 } ( 4 x + 7 ) & 0 \leq x \leq 6 \\ 0 & \text { otherwise } \end{cases}$$ 6

Show that the median of $X$ is 3.87, correct to three significant figures.
[0pt] [3 marks]
6

Find the exact value of $\mathrm { P } ( X > 2 )$

6
The continuous random variable $Y$ has probability density function \(g ( y ) = \begin{cases} \frac { 1 } { 2 } y ^ { 2 } - \frac { 1 } { 6 } y ^ { 3 }	1 \leq y \leq 3
0	\text { otherwise } \end{cases}\)
"
6 (c) (i) Show that $\operatorname { Var } \left( \frac { 1 } { Y } \right) = \frac { 2 } { 81 }$

\multirow[b]{2}{*}{

[4 marks]

}

	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { E }\)
Task \(\boldsymbol { V }\)	100	110	112	102	95
Task \(\boldsymbol { W }\)	125	130	110	120	115
Task \(\boldsymbol { X }\)	105	110	101	108	120
Task \(\boldsymbol { Y }\)	115	115	120	135	110
Task \(\boldsymbol { Z }\)	100	98	99	100	102

Question number	Additional page, if required. Write the question numbers in the left-hand margin.
	\(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

0	12	13	2	0
14	21	0	\(k\)	9
3	10	0	6	23
0	2	6	20	0
0	0	0	0	7

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