Questions - AQA Further Paper 1

Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers $A$ and $B$
3
Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$

AQA Further Paper 1 Specimen Q5

4 marks

5
- 1
3 \end{array} \right] \quad \left[ \begin{array} { l } 2
1
1 \end{array} \right]$$ 2 Use the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$ to show that $\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$
[0pt] [2 marks]
3

Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers $A$ and $B$
3
Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$ 4 A student states that $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos x + \sin x } { \cos x - \sin x } \mathrm {~d} x$ is not an improper integral because $\frac { \cos x + \sin x } { \cos x - \sin x }$ is defined at both $x = 0$ and $x = \frac { \pi } { 2 }$ Assess the validity of the student's argument.
[0pt] [2 marks]
$5 \quad \mathrm { p } ( z ) = z ^ { 4 } + 3 z ^ { 2 } + a z + b , a \in \mathbb { R } , b \in \mathbb { R }$
$2 - 3 \mathrm { i }$ is a root of the equation $\mathrm { p } ( \mathrm { z } ) = 0$ 5
Express $\mathrm { p } ( z )$ as a product of quadratic factors with real coefficients.
5
Solve the equation $\mathrm { p } ( z ) = 0$.

AQA Further Paper 1 Specimen Q6

7 marks

Obtain the general solution of the differential equation $$\tan x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \sin x \tan x$$ where $0 < x < \frac { \pi } { 2 }$
[0pt] [5 marks]
6
Hence find the particular solution of this differential equation, given that $y = \frac { 1 } { 2 \sqrt { 2 } }$
when $x = \frac { \pi } { 4 }$
[0pt] [2 marks]

AQA Further Paper 1 Specimen Q7

5 marks

7 Three planes have equations, $$\begin{gathered} x - y + k z = 3
k x - 3 y + 5 z = - 1
x - 2 y + 3 z = - 4 \end{gathered}$$ Where $k$ is a real constant. The planes do not meet at a unique point. 7

Find the possible values of $k$ 7
There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of $k$ Fully justify your answer.
[0pt] [5 marks]
7
Given further that the equations of the planes form a consistent system, find the solution of the system of equations.

AQA Further Paper 1 Specimen Q8

1 marks

8 A curve has equation $$y = \frac { 5 - 4 x } { 1 + x }$$ 8

Sketch the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-10_1205_1219_886_360} 8
Hence sketch the graph of $y = \left| \frac { 5 - 4 x } { 1 + x } \right|$.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-11_1203_1202_641_331}

AQA Further Paper 1 Specimen Q9

10 marks

9 A line has Cartesian equations $x - p = \frac { y + 2 } { q } = 3 - z$ and a plane has
equation r. $\left[ \begin{array} { r } 1
- 1
- 2 \end{array} \right] = - 3$ 9

In the case where the plane fully contains the line, find the values of $p$ and $q$.
[0pt] [3 marks]
9
In the case where the line intersects the plane at a single point, find the range of values of $p$ and $q$.
[0pt] [3 marks]
9
In the case where the angle $\theta$ between the line and the plane satisfies $\sin \theta = \frac { 1 } { \sqrt { 6 } }$ and the line intersects the plane at $z = 0$ 9
1. Find the value of $q$.
  [0pt] [4 marks]
  9
(ii) Find the value of $p$.

AQA Further Paper 1 Specimen Q10

9 marks

10 The curve, $C$, has equation $y = \frac { x } { \cosh x }$
10

Show that the $x$-coordinates of any stationary points of $C$ satisfy the equation $\tanh x = \frac { 1 } { x }$
[0pt] [3 marks] 10
1. Sketch the graphs of $y = \tanh x$ and $y = \frac { 1 } { x }$ on the axes below.
  [0pt] [2 marks]
  \includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-14_1151_1226_1461_358} 10
(ii) Hence determine the number of stationary points of the curve $C$. 10
Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 0$ at each of the stationary points of the curve $C$.
[0pt] [4 marks]

AQA Further Paper 1 Specimen Q11

6 marks

Prove that $\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } \equiv 2 \operatorname { coth } \theta$ Explicitly state any hyperbolic identities that you use within your proof.
[0pt] [4 marks] LL
LL
LL
LL
LL
LL
LL
L
11
Solve $\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } = 4$ giving your answer in an exact form.
[0pt] [2 marks]

AQA Further Paper 1 Specimen Q12

3 marks

12 The function $\mathrm { f } ( x ) = \cosh ( \mathrm { i } x )$ is defined over the domain $\{ x \in \mathbb { R } : - a \pi \leq x \leq a \pi \}$, where $a$ is a positive integer. By considering the graph of $y = [ f ( x ) ] ^ { n }$, find the mean value of $[ f ( x ) ] ^ { n }$, when $n$ is an odd positive integer. Fully justify your answer.
[0pt] [3 marks]

AQA Further Paper 1 Specimen Q13

5 marks

13 Given that $\mathbf { M } = \left[ \begin{array} { l l l } 1 & 1 & 1
1 & 1 & 1
1 & 1 & 1 \end{array} \right]$, prove that $\mathbf { M } ^ { n } = \left[ \begin{array} { l l l } 3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 }
3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 }
3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 } \end{array} \right]$ for all $n \in \mathbb { N }$
[0pt] [5 marks] LL LL L
LL
LL
LL L
L

AQA Further Paper 1 Specimen Q14

12 marks

14 A particle, $P$, of mass $M$ is released from rest and moves along a horizontal straight line through a point $O$. When $P$ is at a displacement of $x$ metres from $O$, moving with a speed $v \mathrm {~ms} ^ { - 1 }$, a force of magnitude $| 8 M x |$ acts on the particle directed towards $O$. A resistive force, of magnitude $4 M v$, also acts on $P$. 14

Initially $P$ is held at rest at a displacement of 1 metre from $O$. Describe completely the motion of $P$ after it is released. Fully justify your answer.
[0pt] [8 marks]
14
It is decided to alter the resistive force so that the motion of $P$ is critically damped. Determine the magnitude of the resistive force that will produce critically damped motion.
[0pt] [4 marks]

AQA Further Paper 1 Specimen Q15

11 marks

15 An isolated island is populated by rabbits and foxes. At time $t$ the number of rabbits is $x$ and the number of foxes is $y$. It is assumed that:

The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time.
If there were no foxes present, the number of rabbits would increase by $120 \%$ in a unit period of time.
When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to $110 \%$ of the current number of foxes.
At time $t = 0$, the number of foxes is 20 and the number of rabbits is 80 .

1. Construct a mathematical model for the number of rabbits.
  [0pt] [9 marks]
  15
(ii) Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time.
[0pt] [1 mark] 15
Suggest one way in which the model that you have used for the number of rabbits could be refined.
[0pt] [1 mark]

AQA Further Paper 1 2019 June Q1

1 Which one of these functions has the set $\{ x : | x | < 1 \}$ as its greatest possible domain? Circle your answer. $$\cosh x \quad \cosh ^ { - 1 } x \quad \tanh x \quad \tanh ^ { - 1 } x$$

AQA Further Paper 1 2019 June Q2

2 The first two non-zero terms of the Maclaurin series expansion of $\mathrm { f } ( x )$ are $x$ and $- \frac { 1 } { 2 } x ^ { 3 }$ Which one of the following could be $\mathrm { f } ( x )$ ?
Circle your answer.
$x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }$
$\frac { 1 } { 2 } \sin 2 x$
$x \cos x$
$\left( 1 + x ^ { 3 } \right) ^ { - \frac { 1 } { 2 } }$

AQA Further Paper 1 2019 June Q3

3 The function $\mathrm { f } ( x ) = x ^ { 2 } - 1$
Find the mean value of $\mathrm { f } ( x )$ from $x = - 0.5$ to $x = 1.7$
Give your answer to three significant figures.
Circle your answer.

AQA Further Paper 1 2019 June Q4

4 Solve the equation $2 z - 5 \mathrm { i } z ^ { * } = 12$

AQA Further Paper 1 2019 June Q5

5 A plane has equation r. $\left[ \begin{array} { l } 1
1
1 \end{array} \right] = 7$
A line has equation $\mathbf { r } = \left[ \begin{array} { l } 2
0
1 \end{array} \right] + \mu \left[ \begin{array} { l } 1
0
1 \end{array} \right]$
Calculate the acute angle between the line and the plane.
Give your answer to the nearest $0.1 ^ { \circ }$
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-05_2491_1716_219_153}

AQA Further Paper 1 2019 June Q6

Show that $$\cosh ^ { 3 } x + \sinh ^ { 3 } x = \frac { 1 } { 4 } \mathrm { e } ^ { m x } + \frac { 3 } { 4 } \mathrm { e } ^ { n x }$$ where $m$ and $n$ are integers.
6
Hence find $\cosh ^ { 6 } x - \sinh ^ { 6 } x$ in the form $$\frac { a \cosh ( k x ) + b } { 8 }$$ where $a , b$ and $k$ are integers.

AQA Further Paper 1 2019 June Q7

7 Three non-singular square matrices, A, B and $\mathbf { R }$ are such that $$A R = B$$ The matrix $\mathbf { R }$ represents a rotation about the $z$-axis through an angle $\theta$ and $$\mathbf { B } = \left[ \begin{array} { c c c } - \cos \theta & \sin \theta & 0
\sin \theta & \cos \theta & 0
0 & 0 & 1 \end{array} \right]$$ 7

Show that $\mathbf { A }$ is independent of the value of $\theta$.
7
Give a full description of the single transformation represented by the matrix $\mathbf { A }$.

AQA Further Paper 1 2019 June Q8

If $z = \cos \theta + \mathrm { i } \sin \theta$, use de Moivre's theorem to prove that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ 8
Express $\sin ^ { 5 } \theta$ in terms of $\sin 5 \theta , \sin 3 \theta$ and $\sin \theta$
8
Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta d \theta = \frac { 53 } { 480 }$$

AQA Further Paper 1 2019 June Q9

Solve the equation $z ^ { 3 } = \sqrt { 2 } - \sqrt { 6 } \mathrm { i }$, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ where $r > 0$ and $0 \leq \theta < 2 \pi$ 9
The transformation represented by the matrix $\mathbf { M } = \left[ \begin{array} { l l } 5 & 1
1 & 3 \end{array} \right]$ acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points.

AQA Further Paper 1 2019 June Q10

10 The points $A ( 5 , - 4,6 )$ and $B ( 6 , - 6,8 )$ lie on the line $L$. The point $C$ is $( 15 , - 5,9 )$. 10

$D$ is the point on $L$ that is closest to $C$.
Find the coordinates of $D$.
10
Hence find, in exact form, the shortest distance from $C$ to $L$.

AQA Further Paper 1 2019 June Q12

12 Three planes have equations $$\begin{aligned} 4 x - 5 y + z & = 8
3 x + 2 y - k z & = 6
( k - 2 ) x + k y - 8 z & = 6 \end{aligned}$$ where $k$ is a real constant. The planes do not meet at a unique point. 12

Find the possible values of $k$.
12
For each value of $k$ found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system.

AQA Further Paper 1 2019 June Q13

13 The equation $z ^ { 3 } + k z ^ { 2 } + 9 = 0$ has roots $\alpha , \beta$ and $\gamma$. 13

1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$ 13
(ii) Show that $$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$ 13
The equation $9 z ^ { 3 } - 40 z ^ { 2 } + r z + s = 0$ has roots $\alpha \beta + \gamma , \beta \gamma + \alpha$ and $\gamma \alpha + \beta$. 13
1. Show that $$k = - \frac { 40 } { 9 }$$ Question 13 continues on the next page 13
(ii) Without calculating the values of $\alpha , \beta$ and $\gamma$, find the value of $s$. Show working to justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-23_2488_1716_219_153} A light spring is attached to the base of a long tube and has a mass $m$ attached to the other end, as shown in the diagram. The tube is filled with oil. When the compression of the spring is $\varepsilon$ metres, the thrust in the spring is $9 m \varepsilon$ newtons.
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-24_506_250_721_895} The mass is held at rest in a position where the compression of the spring is $\frac { 20 } { 9 }$ metres. The mass is then released from rest. During the subsequent motion the oil causes a resistive force of $6 m v$ newtons to act on the mass, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the mass. At time $t$ seconds after the mass is released, the displacement of the mass above its starting position is $x$ metres.

Questions — AQA Further Paper 1 (97 questions)

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