Questions - AQA - A-Level Maths

AQA Further AS Paper 2 Discrete Specimen Q8

8 marks

8 A family business makes and sells two kinds of kitchen table.
Each pine table takes 6 hours to make and the cost of materials is $\pounds 30$.
Each oak table takes 10 hours to make and the cost of materials is $\pounds 70$.
Each month, the business has 360 hours available for making the tables and $\pounds 2100$ available for the materials.
Each month, the business sells all of its tables to a wholesaler.
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables. Each pine table sold gives the business a profit of $\pounds 40$ and each oak table sold gives the business a profit of $\pounds 75$. Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit. Show clearly how you obtain your answer.
[0pt] [8 marks]

-

T...,

T

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_64_421_1323}

-

T.....

T

\multirow{2}{*}{}

Toloolo

T

To

-

T

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_43_351_660_197}

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_52_208_752_200}

↓

-

- →

Tou

-----

-

T -

-

,

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_27_77_934_1310}

- -

T

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_47_169_898_1567}

T

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_147_950_738}

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_45_261_946_1475}

T

-

.

"
"
"

,

- - -

-

\multirow{4}{*}{}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_29_150_1450_1310}

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_169_1448_1567}

T

-

↓

- -

- - - - - -

-

T -

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_44_183_1637_536}

- -

-

$\%$
- 1
- 1

-

T

- - -

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_35_171_2116_1319}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_41_250_2104_1485}

L

-

- - - -

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_38_158_2143_729}

-

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_45_183_2281_536}

-

- -

-

T

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

6

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

8

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

9

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.

AQA Further Paper 1 2019 June Q14

14

Find $x$ in terms of $t$.
14
State, giving a reason, the type of damping which occurs.

AQA Further Paper 1 2019 June Q15

15 The diagram shows part of a spiral curve. The point $P$ has polar coordinates $( r , \theta )$ where $0 \leq \theta \leq \frac { \pi } { 2 }$
The points $T$ and $S$ lie on the initial line and $O$ is the pole.
$T P Q$ is the tangent to the curve at $P$.
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-26_624_730_653_653} 15

Show that the gradient of $T P Q$ is equal to $$\frac { \frac { \mathrm { d } r } { \mathrm {~d} \theta } \sin \theta + r \cos \theta } { \frac { \mathrm {~d} r } { \mathrm {~d} \theta } \cos \theta - r \sin \theta }$$ 15
The curve has polar equation $$r = \mathrm { e } ^ { ( \cot b ) \theta }$$ where $b$ is a constant such that $0 < b < \frac { \pi } { 2 }$ Use the result of part (a) to show that the angle between the line $O P$ and the tangent TPQ does not depend on $\theta$.
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-28_2488_1719_219_150} Question number Additional page, if required.
Write the question numbers in the left-hand margin.

AQA Further Paper 1 2020 June Q1

1 Which of the integrals below is not an improper integral?
Circle your answer.
$\int _ { 0 } ^ { \infty } e ^ { - x } d x$
$\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x$
$\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x$
$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x$

AQA Further Paper 1 2020 June Q2

1 marks

2 Which one of the matrices below represents a rotation of $90 ^ { \circ }$ about the $x$-axis? Circle your answer.
[0pt] [1 mark]
$\left[ \begin{array} { c c c } 1 & 0 & 0
0 & 1 & 0
0 & 0 & - 1 \end{array} \right]$
$\left[ \begin{array} { c c c } - 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{array} \right]$
$\left[ \begin{array} { l l l } 1 & 0 & 0
0 & 0 & 1
0 & 1 & 0 \end{array} \right]$
$\left[ \begin{array} { c c c } 1 & 0 & 0
0 & 0 & - 1
0 & 1 & 0 \end{array} \right]$

AQA Further Paper 1 2020 June Q3

3 The quadratic equation $a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )$ has real roots $\alpha$ and $\beta$. One of the four statements below is incorrect. Which statement is incorrect? Tick ( $\checkmark$ ) one box.
$c = 0 \Rightarrow \alpha = 0$ or $\beta = 0$ □
$c = a \Rightarrow \alpha$ is the reciprocal of $\beta$ □
$b < 0$ and $c < 0 \Rightarrow \alpha > 0$ and $\beta > 0$ □
$b = 0 \Rightarrow \alpha = - \beta$ □

AQA Further Paper 1 2020 June Q4

4 （a）Express $z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80$ as the product of two quadratic factors with real coefficients．
［4 marks］
4 It is given that $1 - 3 \mathrm { i }$ is one root of the quartic equation
堛的增
4 (b) Find the value of $p$ and the value of $r$.

AQA Further Paper 1 2020 June Q5

4 marks

5

Show that the equation of $H _ { 1 }$ can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where $q$ and $r$ are integers.
5
$\quad \mathrm { H } _ { 2 }$ is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of $H _ { 2 }$ onto the graph of $H _ { 1 }$
[0pt] [4 marks]

AQA Further Paper 1 2020 June Q6

2 marks

6 Let $w$ be the root of the equation $z ^ { 7 } = 1$ that has the smallest argument $\alpha$ in the interval $0 < \alpha < \pi$ 6

Prove that $w ^ { n }$ is also a root of the equation $z ^ { 7 } = 1$ for any integer $n$. 6
Prove that $1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0$
6
Show the positions of $w , w ^ { 2 } , w ^ { 3 } , w ^ { 4 } , w ^ { 5 }$, and $w ^ { 6 }$ on the Argand diagram below.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-08_835_898_1802_571} 6
Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$

AQA Further Paper 1 2020 June Q7

4 marks

7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3
( k - 1 ) x + ( 3 - k ) y + 2 z & = 1
7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7

The planes do not meet at a unique point.
Show that $k = 4.5$ is one possible value of $k$, and find the other possible value of $k$.

7

For each value of $k$ found in part (a), identify the configuration of the given planes.

In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.

[4 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

AQA Further Paper 1 2020 June Q8

8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where $k$ is a constant, form an arithmetic sequence. Find the roots of the equation.

AQA Further Paper 1 2020 June Q9

4 marks

9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9

Find the interval ( $a , b$ ) in which $\mathrm { f } ( x )$ does not take any values.
Fully justify your answer.
9
Find the coordinates of the two stationary points of the graph of $y = \mathrm { f } ( x )$
9
Show that the graph of $y = \mathrm { f } ( x )$ has an oblique asymptote and find its equation.
\section*{Question 9 continues on the next page} 9
Sketch the graph of $y = \mathrm { f } ( x )$ on the axes below.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470}
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
Fird $\begin{aligned} & \text { Do not write }
& \text { outside the } \end{aligned}$

AQA Further Paper 1 2020 June Q10

10

Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
Find the particular solution for which $y = 0$ when $x = 3$
Give your answer in the form $y = \mathrm { f } ( x )$

Questions — AQA (3508 questions)

Browse by module

Browse by board

AQA Further AS Paper 2 Discrete Specimen Q8

AQA Further Paper 1 2019 June Q1

AQA Further Paper 1 2019 June Q2

AQA Further Paper 1 2019 June Q3

AQA Further Paper 1 2019 June Q4

AQA Further Paper 1 2019 June Q5

AQA Further Paper 1 2019 June Q6

AQA Further Paper 1 2019 June Q7

AQA Further Paper 1 2019 June Q8

AQA Further Paper 1 2019 June Q9

AQA Further Paper 1 2019 June Q10

AQA Further Paper 1 2019 June Q12

AQA Further Paper 1 2019 June Q13

AQA Further Paper 1 2019 June Q14

AQA Further Paper 1 2019 June Q15

AQA Further Paper 1 2020 June Q1

AQA Further Paper 1 2020 June Q2

AQA Further Paper 1 2020 June Q3

AQA Further Paper 1 2020 June Q4

AQA Further Paper 1 2020 June Q5

AQA Further Paper 1 2020 June Q6

AQA Further Paper 1 2020 June Q7

AQA Further Paper 1 2020 June Q8

AQA Further Paper 1 2020 June Q9

AQA Further Paper 1 2020 June Q10