Questions - A-Level Maths

Use the definition of the cosh function to prove that $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) = \ln \left( \frac { x + \sqrt { x ^ { 2 } - a ^ { 2 } } } { a } \right) \quad \text { for } a > 0$$ [6 marks]
12
The formulae booklet gives the integral of $\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }$ as $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) \text { or } \ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) + c$$ Ronald says that this contradicts the result given in part (a).
Explain why Ronald is wrong.

AQA Further Paper 1 2020 June Q13

4 marks Standard +0.8

13 Two light elastic strings each have one end attached to a particle $B$ of mass $3 c \mathrm {~kg}$, which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points $A$ and $C$, which are 8 metres apart.
$A B C$ is a horizontal line.
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String $A B$ has a natural length of 4 metres and a stiffness of $5 c$ newtons per metre.
String $B C$ has a natural length of 1 metre and a stiffness of $c$ newtons per metre.
The particle is pulled a distance of $\frac { 1 } { 3 }$ metre from its equilibrium position towards $A$, and released from rest. 13

Show that the particle moves with simple harmonic motion.
13
Find the speed of the particle when it is at a point $P$, a distance $\frac { 1 } { 4 }$ metre from the equilibrium position. Give your answer to two significant figures.
[0pt] [4 marks]

AQA Further Paper 1 2020 June Q14

Challenging +1.2

Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express $\sinh ( m + 1 ) x$ and $\sinh ( m - 1 ) x$ in terms of $\sinh m x , \cosh m x , \sinh x$ and $\cosh x$
14
Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of $\sinh x , \sinh n x$ and $\sinh ( n + 1 ) x$
Do not write
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}

AQA Further Paper 1 2021 June Q1

Easy -1.2

1 Find $$\sum _ { r = 1 } ^ { 20 } \left( r ^ { 2 } - 2 r \right)$$ Circle your answer. 24502660532043680

AQA Further Paper 1 2021 June Q2

Moderate -0.8

2 Given that $z = 1 - 3 \mathrm { i }$ is one root of the equation $z ^ { 2 } + p z + r = 0$, where $p$ and $r$ are real, find the value of $r$. Circle your answer.
$- 8 - 2610$

AQA Further Paper 1 2021 June Q3

Moderate -0.5

3 The curve $C$ has polar equation $$r ^ { 2 } \sin 2 \theta = 4$$ Find a Cartesian equation for $C$.
Circle your answer.
$y = 2 x$
$y = \frac { x } { 2 }$
$y = \frac { 2 } { x }$
$y = 4 x$

AQA Further Paper 1 2021 June Q4

Challenging +1.2

4 Show that the solutions to the equation $$3 \tanh ^ { 2 } x - 2 \operatorname { sech } x = 2$$ can be expressed in the form $$x = \pm \ln ( a + \sqrt { b } )$$ where $a$ and $b$ are integers to be found.
You may use without proof the result $\cosh ^ { - 1 } y = \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right)$

AQA Further Paper 1 2021 June Q5

5 marks Standard +0.8

5 The matrix $\mathbf { M }$ is defined by $\mathbf { M } = \left[ \begin{array} { c c c } 3 & 2 & - 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ Prove by induction that $\mathbf { M } ^ { n } = \left[ \begin{array} { c c c } 3 ^ { n } & 3 ^ { n } - 1 & - 3 ^ { n } + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ for all integers $n \geq 1$ [5 marks]

AQA Further Paper 1 2021 June Q6

Challenging +1.2

Show that the equation $$\left( 2 z - z ^ { * } \right) ^ { * } = z ^ { 2 }$$ has exactly four solutions and state these solutions.
6
1. Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry.
  \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-09_842_860_406_589} 6
(ii) Show that the area of this quadrilateral is $\frac { \sqrt { 15 } } { 2 }$ square units.

AQA Further Paper 1 2021 June Q7

Standard +0.3

7 The diagram below shows the graph of $y = \mathrm { f } ( x ) \quad ( - 4 \leq x \leq 4 )$ The graph meets the $x$-axis at $x = 1$ and $x = 3$
The graph meets the $y$-axis at $y = 2$
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-10_846_854_539_593} 7

Sketch the graph of $y = | \mathrm { f } ( x ) |$ on the axes below.
Show any axis intercepts.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-10_844_844_1601_598} 7
Sketch the graph of $y = \frac { 1 } { \mathrm { f } ( x ) }$ on the axes below.
Show any axis intercepts and asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-11_844_846_495_603} 7
Sketch the graph of $y = \mathrm { f } ( | x | )$ on the axes below.
Show any axis intercepts.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-11_848_849_1647_593}

AQA Further Paper 1 2021 June Q8

Challenging +1.2

8 A particle of mass 4 kg moves horizontally in a straight line. At time $t$ seconds the velocity of the particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$
The following horizontal forces act on the particle:

a constant driving force of magnitude 1.8 newtons
another driving force of magnitude $30 \sqrt { t }$ newtons
a resistive force of magnitude $0.08 v ^ { 2 }$ newtons

When $t = 70 , v = 54$
Use Euler's method with a step length of 0.5 to estimate the velocity of the particle after 71 seconds. Give your answer to four significant figures.

AQA Further Paper 1 2021 June Q9

10 marks Moderate -0.5

9 Use l'Hôpital's rule to show that $$\lim _ { x \rightarrow \infty } \left( x \mathrm { e } ^ { - x } \right) = 0$$ Fully justify your answer.
[0pt] [4 marks]

Evaluate the improper integral \(\int _ { 0 } ^ { 8 } \ln x \mathrm {~d

showing the limiting process.

}

[6 marks]

$11 \quad$ The line $L _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { l } 2 \\ 2 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { c } 2 \\ 3 \\ - 1 \end{array} \right]$
The line $L _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { l } 6 \\ 4 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { c } - 2 \\ 1 \\ 1 \end{array} \right]$

AQA Further Paper 1 2021 June Q11

Challenging +1.2

Find the acute angle between the lines $L _ { 1 }$ and $L _ { 2 }$, giving your answer to the nearest $0.1 ^ { \circ }$
11
The lines $L _ { 1 }$ and $L _ { 2 }$ lie in the plane $\Pi _ { 1 }$
11
1. Find the equation of $\Pi _ { 1 }$, giving your answer in the form r.n $= d$
  11
(ii) Hence find the shortest distance of the plane $\Pi _ { 1 }$ from the origin. 11
The points $A ( 4 , - 1 , - 1 ) , B ( 1,5 , - 7 )$ and $C ( 3,4 , - 8 )$ lie in the plane $\Pi _ { 2 }$
Find the angle between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer to the nearest $0.1 ^ { \circ }$

AQA Further Paper 1 2021 June Q12

4 marks Standard +0.3

12 The matrix $\mathbf { A } = \left[ \begin{array} { c c c } 1 & 5 & 3 \\ 4 & - 2 & p \\ 8 & 5 & - 11 \end{array} \right]$, where $p$ is a constant.
12

Given that $\mathbf { A }$ is a non-singular matrix, find $\mathbf { A } ^ { - 1 }$ in terms of $p$.
State any restrictions on the value of $p$.
12
The equations below represent three planes. $$\begin{aligned} x + 5 y + 3 z & = 5 \\ 4 x - 2 y + p z & = 24 \\ 8 x + 5 y - 11 z & = - 30 \end{aligned}$$ 12
1. Find, in terms of $p$, the coordinates of the point of intersection of the three planes.
  [0pt] [4 marks]
  12
(ii) In the case where $p = 2$, show that the planes are mutually perpendicular.

AQA Further Paper 1 2021 June Q13

Standard +0.8

13
The transformation S is represented by the matrix $\left[ \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right]$
The transformation T is a translation by the vector $\left[ \begin{array} { c } 0 \\ - 5 \end{array} \right]$
Kamla transforms the graphs of various functions by applying first S , then T .
Leo says that, for some graphs, Kamla would get a different result if she applied first $T$, then $S$. Kamla disagrees.
State who is correct.
Fully justify your answer.

AQA Further Paper 1 2021 June Q14

Challenging +1.8

14 The hyperbola $H$ has equation $y ^ { 2 } - x ^ { 2 } = 16$ The circle $C$ has equation $x ^ { 2 } + y ^ { 2 } = 32$
The diagram below shows part of the graph of $H$ and part of the graph of $C$.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-22_825_716_539_662} Show that the shaded region in the first quadrant enclosed by $H , C$, the $x$-axis and the $y$-axis has area $$\frac { 16 \pi } { 3 } + 8 \ln \left( \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 } \right)$$

AQA Further Paper 1 2021 June Q15

Challenging +1.2

15 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A particle $P$ of mass $m$ is attached to two light elastic strings, $A P$ and $B P$.
The other ends of the strings, $A$ and $B$, are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container. The coefficient of friction between $P$ and the surface is 0.68

When the extension of string $A P$ is $e _ { A }$ metres, the tension in $A P$ is $24 m e _ { A }$
When the extension of string $B P$ is $e _ { B }$ metres, the tension in $B P$ is $10 m e _ { B }$
The natural length of string $A P$ is 1 metre
The natural length of string $B P$ is 1.3 metres
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-24_92_1082_1030_479}

Show that when $A P = 1.5$ metres, the tension in $A P$ is equal to the tension in $B P$.
15
$\quad P$ is held at the point between $A$ and $B$ where $A P = 1.9$ metres, and then released from rest. At time $t$ seconds after $P$ is released, $A P = ( 1.5 + x )$ metres.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-25_140_1068_493_484} Show that when $P$ is moving towards $A$, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 34 x = 6.664$$ 15
The container is then filled with oil, and $P$ is again released from rest at the point between $A$ and $B$ where $A P = 1.9$ metres. At time $t$ seconds after $P$ is released, the oil causes a resistive force of magnitude $10 m v$ newtons to act on the particle, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the particle. Find $x$ in terms of $t$ when $P$ is moving towards $A$.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-27_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-32_2486_1719_221_150}

AQA Further Paper 1 2022 June Q1

1 marks Moderate -0.5

1 The displacement of a particle from its equilibrium position is $x$ metres at time $t$ seconds. The motion of the particle obeys the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 9 x$$ Calculate the period of its motion in seconds.
Circle your answer.
[0pt] [1 mark]
$\frac { \pi } { 9 }$
$\frac { 2 \pi } { 9 }$
$\frac { \pi } { 3 }$
$\frac { 2 \pi } { 3 }$

AQA Further Paper 1 2022 June Q2

Moderate -0.5

AQA Further Paper 1 2022 June Q4

1 marks Standard +0.3

4 The vector $\mathbf { v }$ is an eigenvector of the matrix $\mathbf { N }$ with corresponding eigenvalue 4
The vector $\mathbf { v }$ is also an eigenvector of the matrix $\mathbf { M }$ with corresponding eigenvalue 3
Given that $$\mathbf { N M } ^ { 2 } \mathbf { v } = \lambda \mathbf { v }$$ find the value of $\lambda$
Circle your answer.
[0pt] [1 mark]
102436144

AQA Further Paper 1 2022 June Q5

Standard +0.8

5 It is given that $z = - \frac { 3 } { 2 } + \mathrm { i } \frac { \sqrt { 11 } } { 2 }$ is a root of the equation $$z ^ { 4 } - 3 z ^ { 3 } - 5 z ^ { 2 } + k z + 40 = 0$$ where $k$ is a real number.
5

Find the other three roots.
5
Given that $x \in \mathbb { R }$, solve $$x ^ { 4 } - 3 x ^ { 3 } - 5 x ^ { 2 } + k x + 40 < 0$$

AQA Further Paper 1 2022 June Q6

Standard +0.8

Given that $| x | < 1$, prove that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$$ 6
Solve the equation $$20 \operatorname { sech } ^ { 2 } x - 11 \tanh x = 16$$ Give your answer in logarithmic form.
$7 \quad$ The matrix $\mathbf { M }$ is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 7 & - 3 \\ 3 & 6 & k + 1 \\ 1 & 3 & 2 \end{array} \right]$$ where $k$ is a constant.

AQA Further Paper 1 2022 June Q7

Standard +0.3

1. Given that $\mathbf { M }$ is a non-singular matrix, find $\mathbf { M } ^ { - 1 }$ in terms of $k$
  7
(ii) State any restrictions on the value of $k$ 7
Using your answer to part (a)(i), solve $$\begin{array} { r } x + 7 y - 3 z = 6 \\ 3 x + 6 y + 6 z = 3 \\ x + 3 y + 2 z = 1 \end{array}$$

AQA Further Paper 1 2022 June Q8

Challenging +1.2

The complex number $w$ is such that $$\arg ( w + 2 \mathrm { i } ) = \tan ^ { - 1 } \frac { 1 } { 2 }$$ It is given that $w = x + \mathrm { i } y$, where $x$ and $y$ are real and $x > 0$
Find an equation for $y$ in terms of $x$
8
The complex number $z$ satisfies both $$- \frac { \pi } { 2 } \leq \arg ( z + 2 \mathrm { i } ) \leq \tan ^ { - 1 } \frac { 1 } { 2 } \quad \text { and } \quad | z - 2 + 3 \mathrm { i } | \leq 2$$ The region $R$ is the locus of $z$
Sketch the region $R$ on the Argand diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-10_1015_1020_1683_511} 8
$\quad z _ { 1 }$ is the point in $R$ at which $| z |$ is minimum. 8
1. Calculate the exact value of $\left| z _ { 1 } \right|$
  8
(ii) Express $z _ { 1 }$ in the form $a + \mathrm { i } b$, where $a$ and $b$ are real.

AQA Further Paper 1 2022 June Q9

4 marks

9 Roberto is solving this mathematics problem: The curve $C _ { 1 }$ has polar equation $$r ^ { 2 } = 9 \sin 2 \theta$$ for all possible values of $\theta$
Find the area enclosed by $C _ { 1 }$ Roberto's solution is as follows: $$\begin{aligned} A & = \frac { 1 } { 2 } \int _ { - \pi } ^ { \pi } 9 \sin 2 \theta \mathrm {~d} \theta \\ & = \left[ - \frac { 9 } { 4 } \cos 2 \theta \right] _ { - \pi } ^ { \pi } \\ & = 0 \end{aligned}$$ 9

$\quad$ Sketch the curve $C _ { 1 }$ 9
Explain what Roberto has done wrong.
9
$\quad$ Find the area enclosed by $C _ { 1 }$
9
$\quad P$ and $Q$ are distinct points on $C _ { 1 }$ for which $r$ is a maximum. $P$ is above the initial line. Find the polar coordinates of $P$ and $Q$
9
The matrix $\mathbf { M } = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]$ represents the transformation T T maps $C _ { 1 }$ onto a curve $C _ { 2 }$
9
1. T maps $P$ onto the point $P ^ { \prime }$
  Find the polar coordinates of $P ^ { \prime }$
  [0pt] [4 marks]
  9
(ii) Find the area enclosed by $C _ { 2 }$ Fully justify your answer.

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