Questions Further Paper 2

Find a sequence of transformations that maps the graph of $C _ { 1 }$ onto the graph of $C _ { 2 }$ [4 marks]
10
Find the equations of the asymptotes to $C _ { 2 }$
Give your answers in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers.

AQA Further Paper 2 2022 June Q11

2 marks

Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 5 } { 2 } & - \frac { 3 } { 2 }
- \frac { 3 } { 2 } & \frac { 13 } { 2 } \end{array} \right]$$ 11
1. Describe how the directions of the invariant lines of the transformation represented by $\mathbf { M }$ are related to each other. Fully justify your answer.
  [0pt] [2 marks]
  11
(ii) Describe fully the transformation represented by $\mathbf { M }$

AQA Further Paper 2 2022 June Q12

12 The shaded region shown in the diagram below is bounded by the $x$-axis, the curve $y = \mathrm { f } ( x )$, and the lines $x = a$ and $x = b$
\includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-16_661_721_406_662} The shaded region is rotated through $2 \pi$ radians about the $x$-axis to form a solid.
12

Show that the volume of this solid is $$\pi \int _ { a } ^ { b } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x$$ 12
In the case where $a = 1 , b = 2$ and $$f ( x ) = \frac { x + 3 } { ( x + 1 ) \sqrt { x } }$$ show that the volume of the solid is $$\pi \left( \ln \left( \frac { 2 ^ { m } } { 3 ^ { n } } \right) - \frac { 2 } { 3 } \right)$$ where $m$ and $n$ are integers.

AQA Further Paper 2 2022 June Q13

4 marks

The matrix A represents a reflection in the line $y = m x$, where $m$ is a constant. Show that $\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m
2 m & m ^ { 2 } - 1 \end{array} \right]$
You may use the result in the formulae booklet. 13
$\quad$ The matrix $\mathbf { B }$ is defined as $\mathbf { B } = \left[ \begin{array} { l l } 3 & 0
0 & 3 \end{array} \right]$
Show that $( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }$
where $\mathbf { I }$ is the $2 \times 2$ identity matrix and $k$ is an integer.
13
1. The diagram below shows a point $P$ and the line $y = m x$ Draw four lines on the diagram to demonstrate the result proved in part (b).
  Label as $P ^ { \prime }$ the image of $P$ under the transformation represented by (BA) ${ } ^ { 2 }$
  \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13
(ii) Explain how your completed diagram shows the result proved in part (b).
13
The matrix $\mathbf { C }$ is defined as $\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 }
\frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]$
Find the value of $m$ such that $\mathbf { C } = \mathbf { B A }$ Fully justify your answer.
[0pt] [4 marks]

AQA Further Paper 2 2022 June Q14

5 marks

14 On an isolated island some rabbits have been accidently introduced. In order to eliminate them, conservationists have introduced some birds of prey.
At time $t$ years $( t \geq 0 )$ there are $x$ rabbits and $y$ birds of prey.
At time $t = 0$ there are 1755 rabbits and 30 birds of prey.
When $t > 0$ it is assumed that:

the rabbits will reproduce at a rate of $a \%$ per year
each bird of prey will kill, on average, $b$ rabbits per year
the death rate of the birds of prey is $c$ birds per year
the number of birds of prey will increase at a rate of $d \%$ of the rabbit population per year.

This system is represented by the coupled differential equations: $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4 x - 13 y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.01 x - 1.95 \end{aligned}$$ 14

State the value of $a$, the value of $b$, the value of $c$ and the value of $d$
[0pt] [2 marks]
14
Solve the coupled differential equations to find both $x$ and $y$ in terms of $t$

AQA Further Paper 2 2023 June Q1

1 Given that $y = \sin x + \sinh x$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y$
Circle your answer. $$\begin{array} { l l l l } 2 \sin x & - 2 \sin x & 2 \sinh x & - 2 \sinh x \end{array}$$

AQA Further Paper 2 2023 June Q2

1 marks

2 Which one of the expressions below is not equal to zero?
Circle your answer.
[0pt] [1 mark]
$\lim _ { x \rightarrow \infty } \left( x ^ { 2 } \mathrm { e } ^ { - x } \right)$
$\lim _ { x \rightarrow 0 } \left( x ^ { 5 } \ln x \right)$
$\lim _ { x \rightarrow \infty } \left( \frac { \mathrm { e } ^ { x } } { x ^ { 5 } } \right)$
$\lim _ { x \rightarrow 0 } \left( x ^ { 3 } \mathrm { e } ^ { x } \right)$

AQA Further Paper 2 2023 June Q3

3 The determinant $A = \left| \begin{array} { l l l } 1 & 1 & 1
2 & 0 & 2
3 & 2 & 1 \end{array} \right|$
Which one of the determinants below has a value which is not equal to the value of $A$ ?
Tick ( $\checkmark$ ) one box.

\(\left\| \begin{array} { l l l } 3	1	3
2	0	2
3	2	1 \end{array} \right\|\)	$\square$
\(\left\| \begin{array} { l l l } 1	2	3
1	0	2
1	2	1 \end{array} \right\|\)	$\square$
\(\left\| \begin{array} { l l l } 2	2	2
1	0	1
3	2	1 \end{array} \right\|\)	$\square$
\(\left\| \begin{array} { l l l } 1	1	1
3	2	1
2	0	2 \end{array} \right\|\)	$\square$

AQA Further Paper 2 2023 June Q4

1 marks

4 It is given that $\mathrm { f } ( x ) = \cosh ^ { - 1 } ( x - 3 )$
Which of the sets listed below is the greatest possible domain of the function $f$ ?
Circle your answer.
[0pt] [1 mark] $$\{ x : x \geq 4 \} \quad \{ x : x \geq 3 \} \quad \{ x : x \geq 1 \} \quad \{ x : x \geq 0 \}$$

AQA Further Paper 2 2023 June Q5

5 Josh and Zoe are solving the following mathematics problem: The curve $C _ { 1 }$ has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The matrix $\mathbf { M } = \left[ \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right]$ maps $C _ { 1 }$ onto $C _ { 2 }$
Find the equations of the asymptotes of $C _ { 2 }$ Josh says that to solve this problem you must first carry out the transformation on $C _ { 1 }$ to find $C _ { 2 }$, and then find the asymptotes of $C _ { 2 }$ Zoe says that you will get the same answer if you first find the asymptotes of $C _ { 1 }$, and then carry out the transformation on these asymptotes to obtain the asymptotes of $C _ { 2 }$ Show that Zoe is correct.

AQA Further Paper 2 2023 June Q6

Express $- 5 - 5 \mathrm { i }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $- \pi < \theta \leq \pi$
6
The point on an Argand diagram that represents $- 5 - 5 \mathrm { i }$ is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle.
Give your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $- \pi < \theta \leq \pi$

AQA Further Paper 2 2023 June Q7

7 Show that $$\sum _ { r = 11 } ^ { n + 1 } r ^ { 3 } = \frac { 1 } { 4 } \left( n ^ { 2 } + a n + b \right) \left( n ^ { 2 } + a n + c \right)$$ where $a$, $b$ and $c$ are integers to be found.
$8 \quad \mathbf { A }$ is a non-singular $2 \times 2$ matrix and $\mathbf { A } ^ { \mathrm { T } }$ is the transpose of $\mathbf { A }$

AQA Further Paper 2 2023 June Q8

Using the result $$( \mathbf { A B } ) ^ { \mathrm { T } } = \mathbf { B } ^ { \mathrm { T } } \mathbf { A } ^ { \mathrm { T } }$$ show that $$\left( \mathbf { A } ^ { - 1 } \right) ^ { \mathrm { T } } = \left( \mathbf { A } ^ { \mathrm { T } } \right) ^ { - 1 }$$ 8
It is given that $\mathbf { A } = \left[ \begin{array} { c c } 4 & 5
- 1 & k \end{array} \right]$, where $k$ is a real constant.
8
1. Find $\left( \mathbf { A } ^ { - 1 } \right) ^ { \mathrm { T } }$, giving your answer in terms of $k$
  8
(ii) State the restriction on the possible values of $k$

AQA Further Paper 2 2023 June Q9

9 The complex number $z$ is such that $$z = \frac { 1 + \mathrm { i } } { 1 - k \mathrm { i } }$$ where $k$ is a real number. 9

Find the real part of $z$ and the imaginary part of $z$, giving your answers in terms of $k$
9
In the case where $k = \sqrt { 3 }$, use part (a) to show that $$\cos \frac { 7 \pi } { 12 } = \frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$$ $\_\_\_\_$ The region $R$ on an Argand diagram satisfies both $| z + 2 \mathrm { i } | \leq 3$ and $- \frac { \pi } { 6 } \leq \arg ( z ) \leq \frac { \pi } { 2 }$

AQA Further Paper 2 2023 June Q10

3 marks

Sketch $R$ on the Argand diagram below.
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}, center]{bc1b33a7-800b-4359-b7ba-6460f17984e5-10_1205_1200_520_422} 10
Find the maximum value of $| z |$ in the region $R$, giving your answer in exact form.

AQA Further Paper 2 2023 June Q11

11 The line $l _ { 1 }$ passes through the points $A ( 6,2,7 )$ and $B ( 4 , - 3,7 )$ 11

Find a Cartesian equation of $l _ { 1 }$
11
The line $l _ { 2 }$ has vector equation $\mathbf { r } = \left[ \begin{array} { l } 8
9
c \end{array} \right] + \mu \left[ \begin{array} { l } 1
1
2 \end{array} \right]$ where $c$ is a constant.
11
1. Explain how you know that the lines $l _ { 1 }$ and $l _ { 2 }$ are not perpendicular.
  11
(ii) The lines $l _ { 1 }$ and $l _ { 2 }$ both lie in the same plane. Find the value of $c$

AQA Further Paper 2 2023 June Q13

9 marks

13 The quadratic equation $z ^ { 2 } - 5 z + 8 = 0$ has roots $\alpha$ and $\beta$ 13

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$
13
Without finding the value of $\alpha$ or the value of $\beta$, show that $\alpha ^ { 4 } + \beta ^ { 4 } = - 47$
[0pt] [4 marks]

\includegraphics[max width=\textwidth, alt={}]{bc1b33a7-800b-4359-b7ba-6460f17984e5-18_2495_1917_212_150}

AQA Further Paper 2 2023 June Q14

14 (c) Find the value of $\int _ { - 2 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x$
Fully justify your answer.

AQA Further Paper 2 2023 June Q15

5 marks

Given that $z = \cos \theta + \mathrm { i } \sin \theta$, use de Moivre's theorem to show that $$z ^ { n } - z ^ { - n } = 2 i \sin n \theta$$ 15
The series $S$ is defined as $$S = \sin \theta + \sin 3 \theta + \ldots + \sin ( 2 n - 1 ) \theta$$ Use part (a) to express $S$ in the form $$S = \frac { 1 } { 2 \mathrm { i } } \left( G _ { 1 } \right) - \frac { 1 } { 2 \mathrm { i } } \left( G _ { 2 } \right)$$ where each of $G _ { 1 }$ and $G _ { 2 }$ is a geometric series.

15
Hence, show that [5 marks]

AQA Further Paper 2 2023 June Q16

6 marks

16 A bungee jumper of mass $m \mathrm {~kg}$ is attached to an elastic rope.
The other end of the rope is attached to a fixed point.
The bungee jumper falls vertically from the fixed point.
At time $t$ seconds after the rope first becomes taut, the extension of the rope is $x$ metres and the speed of the bungee jumper is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ 16

A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are
- the weight of the bungee jumper
- a tension in the rope of magnitude $k x$ newtons
- an air resistance force of magnitude $R v$ newtons
  where $k$ and $R$ are constants such that $4 k m > R ^ { 2 }$
  16
- 1. Show that this model gives the result
$$\left. \left. x = \mathrm { e } ^ { - \frac { R t } { 2 m } } \left( A \cos \frac { \sqrt { 4 k m - R ^ { 2 } } } { 2 m } \right) t + B \sin \frac { \sqrt { 4 k m - R ^ { 2 } } } { 2 m } \right) t \right) + \frac { m g } { k }$$ where $A$ and $B$ are constants, and $g \mathrm {~ms} ^ { - 2 }$ is the acceleration due to gravity.
You do not need to find the value of $A$ or the value of $B$
16
(ii) It is also given that: $$\begin{aligned} k & = 16
R & = 20
m & = 62.5
g & = 9.8 \mathrm {~ms} ^ { - 2 } \end{aligned}$$ and that the speed of the bungee jumper when the rope becomes taut is $14 \mathrm {~ms} ^ { - 1 }$ Show that, to the nearest integer, $A = - 38$ and $B = 16$
[0pt] [6 marks]
16
A second, simpler model assumes that the air resistance is zero. The values of $k , m$ and $g$ remain the same.
Find an expression for $x$ in terms of $t$ according to this simpler model, giving the values of all constants to two significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{bc1b33a7-800b-4359-b7ba-6460f17984e5-26_2488_1719_219_150}

AQA Further Paper 2 2024 June Q1

1 It is given that $$\left[ \begin{array} { l } 2
1
3 \end{array} \right] \cdot \left[ \begin{array} { c } 5
\lambda
- 6 \end{array} \right] = 0$$ where $\lambda$ is a constant. Find the value of $\lambda$ Circle your answer. -28-8 828

AQA Further Paper 2 2024 June Q2

1 marks

2 The movement of a particle is described by the simple harmonic equation $$\ddot { x } = - 25 x$$ where $x$ metres is the displacement of the particle at time $t$ seconds, and $\ddot { x } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ is the acceleration of the particle. The maximum displacement of the particle is 9 metres. Find the maximum speed of the particle.
Circle your answer.
[0pt] [1 mark]
$15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
$45 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
$75 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
$135 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

AQA Further Paper 2 2024 June Q3

3 The function $g$ is defined by $$g ( x ) = \operatorname { sech } x \quad ( x \in \mathbb { R } )$$ Which one of the following is the range of $g$ ?
Tick $( \checkmark )$ one box.
$- \infty < \mathrm { g } ( x ) \leq - 1$
\includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-03_117_117_635_854}
$- 1 \leq \mathrm { g } ( x ) < 0$
\includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-03_113_113_785_854}
$0 < \mathrm { g } ( x ) \leq 1$
\includegraphics[max width=\textwidth, alt={}, center]{99b03f18-6dd6-437d-8b01-009ca7ab49ea-03_117_117_927_854}
$1 \leq g ( x ) \leq \infty$ □

AQA Further Paper 2 2024 June Q4

4 The function f is a quartic function with real coefficients.
The complex number 5i is a root of the equation $\mathrm { f } ( x ) = 0$
Which one of the following must be a factor of $\mathrm { f } ( x )$ ?
Circle your answer.
( $x ^ { 2 } - 25$ )
$\left( x ^ { 2 } - 5 \right)$
$\left( x ^ { 2 } + 5 \right)$
$\left( x ^ { 2 } + 25 \right)$

AQA Further Paper 2 2024 June Q5

5 The first four terms of the series $S$ can be written as $$S = ( 1 \times 2 ) + ( 2 \times 3 ) + ( 3 \times 4 ) + ( 4 \times 5 ) + \ldots$$ 5

Write an expression, using $\sum$ notation, for the sum of the first $n$ terms of $S$ 5
Show that the sum of the first $n$ terms of $S$ is equal to $$\frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$

\(\left\| \begin{array} { l l l } 3	1	3
2	0	2
3	2	1 \end{array} \right\|\)	\(\square\)
\(\left\| \begin{array} { l l l } 1	2	3
1	0	2
1	2	1 \end{array} \right\|\)	\(\square\)
\(\left\| \begin{array} { l l l } 2	2	2
1	0	1
3	2	1 \end{array} \right\|\)	\(\square\)
\(\left\| \begin{array} { l l l } 1	1	1
3	2	1
2	0	2 \end{array} \right\|\)	\(\square\)

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