Questions - AQA - A-Level Maths

It is given that Starting from the exponential definition of the sinh function, show that $\sinh p = r$ $$p = \ln \left( r + \sqrt { r ^ { 2 } + 1 } \right)$$ Staring fr
9
Solve the equation $$\cosh ^ { 2 } x = 2 \sinh x + 16$$ Give your answers in logarithmic form.
[0pt] [4 marks]
The complex numbers $z$ and $w$ are defined by $$\begin{aligned} z & = \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 }
\text { and } \quad w & = \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \end{aligned}$$ By evaluating the product $z w$, show that $$\tan \frac { 5 \pi } { 12 } = 2 + \sqrt { 3 }$$

AQA Further Paper 1 2024 June Q11

Find $\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 2 } \tan ^ { - 1 } x \right)$ 11
Hence find $\int 2 x \tan ^ { - 1 } x \mathrm {~d} x$

AQA Further Paper 1 2024 June Q12

4 marks

12 The line $L _ { 1 }$ has equation $$\mathbf { r } = \left[ \begin{array} { l } 4
2
1 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
3
- 1 \end{array} \right]$$ The transformation T is represented by the matrix $$\left[ \begin{array} { c c c } 2 & 1 & 0
3 & 4 & 6
- 5 & 2 & - 3 \end{array} \right]$$ The transformation T transforms the line $L _ { 1 }$ to the line $L _ { 2 }$ 12

Show that the angle between $L _ { 1 }$ and $L _ { 2 }$ is 0.701 radians, correct to three decimal places.
[0pt] [4 marks]

12
Find the shortest distance between $L _ { 1 }$ and $L _ { 2 }$
Give your answer in an exact form.

AQA Further Paper 1 2024 June Q13

Use de Moivre's theorem to show that $$\cos 3 \theta = 4 \cos ^ { 3 } \theta - 3 \cos \theta$$ 13
Use de Moivre's theorem to express $\sin 3 \theta$ in terms of $\sin \theta$
13
Hence show that $$\cot 3 \theta = \frac { \cot ^ { 3 } \theta - 3 \cot \theta } { 3 \cot ^ { 2 } \theta - 1 }$$

AQA Further Paper 1 2024 June Q14

14 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tanh x = \sinh ^ { 3 } x$$ given that $y = 3$ when $x = \ln 2$
Give your answer in an exact form.

AQA Further Paper 1 2024 June Q15

15 A curve is defined parametrically by the equations $$\begin{array} { l l } x = \frac { 3 } { 2 } t ^ { 3 } + 5 &
y = t ^ { \frac { 9 } { 2 } } & ( t \geq 0 ) \end{array}$$ Show that the arc length of the curve from $t = 0$ to $t = 2$ is equal to 26 units.

AQA Further Paper 1 2024 June Q16

16 The curve $C$ has polar equation $r = 2 + \tan \theta$ The curve $C$ meets the line $\theta = \frac { \pi } { 4 }$ at the point $A$
The point $B$ has polar coordinates $( 4,0 )$
The diagram shows part of the curve $C$, and the points $A$ and $B$
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-22_515_1168_575_427} 16

Show that the area of triangle $O A B$ is $3 \sqrt { 2 }$ units.
16
Find the area of the shaded region.
Give your answer in an exact form.

AQA Further Paper 1 2024 June Q17

17 By making a suitable substitution, show that $$\int _ { - 2 } ^ { 1 } \sqrt { x ^ { 2 } + 6 x + 8 } d x = 2 \sqrt { 15 } - \frac { 1 } { 2 } \cosh ^ { - 1 } ( 4 )$$

AQA Further Paper 1 2024 June Q18

3 marks

18 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ Two light elastic strings each have one end attached to a small ball $B$ of mass 0.5 kg The other ends of the strings are attached to the fixed points $A$ and $C$, which are 8 metres apart with $A$ vertically above $C$ The whole system is in a thin tube of oil, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-26_439_154_685_927} The string connecting $A$ and $B$ has natural length 2 metres, and the tension in this string is $7 e$ newtons when the extension is $e$ metres. The string connecting $B$ and $C$ has natural length 3 metres, and the tension in this string is $3 e$ newtons when the extension is $e$ metres. 18

Find the extension of each string when the system is in equilibrium.
18
It is known that in a large bath of oil, the oil causes a resistive force of magnitude $4.5 v$ newtons to act on the ball, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii). 18
1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards C, and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 9 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 20 x = 0$$ where $x$ metres is the displacement of the particle below the equilibrium position at time $t$ seconds after the particle is released.
  [0pt] [3 marks]
  18
(ii) Find $x$ in terms of $t$
29 18
State one limitation of the model used in part (b)

AQA Further Paper 2 2019 June Q1

1 marks

1 Given that $z$ is a complex number, and that $z ^ { * }$ is the complex conjugate of $z$, which of the following statements is not always true? Circle your answer.
[0pt] [1 mark] $$\left( z ^ { * } \right) ^ { * } = z \quad z z ^ { * } = | z | ^ { 2 } \quad ( - z ) ^ { * } = - \left( z ^ { * } \right) \quad z - z ^ { * } = z ^ { * } - z$$

AQA Further Paper 2 2019 June Q2

1 marks

2 Which of the straight lines given below is an asymptote to the curve $$y = \frac { a x ^ { 2 } } { x - 1 }$$ where $a$ is a non-zero constant? Circle your answer.
[0pt] [1 mark]
$y = a x + a$
$y = a x$
$y = a x - a$
$y = a$

AQA Further Paper 2 2019 June Q3

1 marks

3 The set $\mathcal { A }$ is defined by $\mathcal { A } = \{ x : - \sqrt { } 2 < x < 0 \} \cup \{ x : 0 < x < \sqrt { } 2 \}$
Which of the inequalities given below has $\mathcal { A }$ as its solution?
Circle your answer.
[0pt] [1 mark]
$\left| x ^ { 2 } - 1 \right| > 1$
$\left| x ^ { 2 } - 1 \right| \geq 1$
$\left| x ^ { 2 } - 1 \right| < 1$
$\left| x ^ { 2 } - 1 \right| \leq 1$

AQA Further Paper 2 2019 June Q4

4 The positive integer $k$ is such that $$\sum _ { r = 1 } ^ { k } ( 3 r - k ) = 90$$ Find the value of $k$.

AQA Further Paper 2 2019 June Q5

5 A curve has equation $y = \cosh x$ Show that the arc length of the curve from $x = a$ to $x = b$, where $0 < a < b$, is equal to
$\sinh b - \sinh a$
$6 \quad$ A circle $C$ in the complex plane has equation $| z - 2 - 5 \mathrm { i } | = a$ The point $z _ { 1 }$ on $C$ has the least argument of any point on $C$, and $\arg \left( z _ { 1 } \right) = \frac { \pi } { 4 }$ Prove that $a = \frac { 3 \sqrt { } 2 } { 2 }$

AQA Further Paper 2 2019 June Q7

4 marks

7 The points $A , B$ and $C$ have coordinates $A ( 4,5,2 ) , B ( - 3,2 , - 4 )$ and $C ( 2,6,1 )$ 7

Use a vector product to show that the area of triangle $A B C$ is $\frac { 5 \sqrt { 11 } } { 2 }$
[0pt] [4 marks]
7
The points $A , B$ and $C$ lie in a plane.
Find a vector equation of the plane in the form r.n $= k$
7
Hence find the exact distance of the plane from the origin.

AQA Further Paper 2 2019 June Q8

The line $y = m x$ is a tangent to $P _ { 2 }$
Prove that $m = \pm \sqrt { \frac { a } { b } }$
Solutions using differentiation will be given no marks.
8
The line $y = \sqrt { \frac { a } { b } } x$ meets $P _ { 2 }$ at the point $D$.
The finite region $R$ is bounded by the $x$-axis, $P _ { 2 }$ and a line through $D$ perpendicular to the $x$-axis. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid.
Find, in terms of $a$ and $b$, the volume of this solid.
Fully justify your answer.
Find the eigenvalues and corresponding eigenvectors of the matrix

AQA Further Paper 2 2019 June Q9

4 marks

Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { 5 } & \frac { 2 } { 5 }
\frac { - 3 } { 5 } & \frac { 13 } { 10 } \end{array} \right]$$ 9
Find matrices $\mathbf { U }$ and $\mathbf { D }$ such that $\mathbf { D }$ is a diagonal matrix and $\mathbf { M } = \mathbf { U D U } ^ { - 1 }$
9
Given that $\mathbf { M } ^ { n } \rightarrow \mathbf { L }$ as $n \rightarrow \infty$, find the matrix $\mathbf { L }$.
[0pt] [4 marks]
9
The transformation represented by $\mathbf { L }$ maps all points onto a line. Find the equation of this line.
\begin{center} \begin{tabular}{ | l | }

AQA Further Paper 2 2019 June Q10

7 marks

10
- $\begin{array} { c } \text { Prove by induction that } \mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n \text { is divisible by } 6 \text { for all integers } n \geq 1
\text { [7 marks] }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - } \end{array}$
-
\end{tabular} \end{center}

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-13_2488_1716_219_153}

AQA Further Paper 2 2019 June Q11

11 The line $L _ { 1 }$ has equation $$\frac { x - 2 } { 3 } = \frac { y + 4 } { 8 } = \frac { 4 z - 5 } { 5 }$$ The line $L _ { 2 }$ has equation $$\left( \mathbf { r } - \left[ \begin{array} { c } - 2
0
3 \end{array} \right] \right) \times \left[ \begin{array} { l } 2
1
3 \end{array} \right] = \mathbf { 0 }$$ Find the shortest distance between the two lines, giving your answer to three significant figures.

AQA Further Paper 2 2019 June Q12

10 marks

12 Abel and Bonnie are trying to solve this mathematical problem: $$\begin{gathered} z = 2 - 3 \mathrm { i } \text { is a root of the equation }
2 z ^ { 3 } + m z ^ { 2 } + p z + 91 = 0 \end{gathered}$$ Find the value of $m$ and the value of $p$. Abel says he has solved the problem.
Bonnie says there is not enough information to solve the problem.
12

Abel's solution begins as follows: Since $z = 2 - 3 \mathrm { i }$ is a root of the equation, $z = 2 + 3 \mathrm { i }$ is another root. State one extra piece of information about $m$ and $p$ which could be added to the problem to make the beginning of Abel's solution correct.

12	Prove that Bonnie is right.

13(a) Explain why $\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$ is an improper integral.

[1 mark]

13(b) Evaluate $\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$

Show the limiting process.

[9 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-18_97_150_215_1884}

AQA Further Paper 2 2019 June Q14

Use the method of differences to show that $$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$ where $a , b$ and $c$ are integers.
Question 14 continues on the next page 14
Show that, for any number $k$ greater than $\frac { 12 } { 5 }$, if the difference between $\frac { 5 } { 12 }$ and $S _ { n }$ is less than $\frac { 1 } { k }$, then $$n > \frac { k - 5 + \sqrt { k ^ { 2 } + 1 } } { 2 }$$

AQA Further Paper 2 2019 June Q15

9 marks

Find the value of $r$. 15
Show that $\mu = 3$ 15
Solve the coupled differential equations to find both $x$ and $y$ in terms of $t$.
[0pt] [9 marks]

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-27_2493_1721_214_150}
Additional page, if required.
Write the question numbers in the left-hand margin.

AQA Further Paper 2 2020 June Q1

1 Three of the four expressions below are equivalent to each other.
Which of the four expressions is not equivalent to any of the others? Circle your answer.
$\mathbf { a } \times ( \mathbf { a } + \mathbf { b } )$
$( \mathbf { a } + \mathbf { b } ) \times \mathbf { b }$
$( \mathbf { a } - \mathbf { b } ) \times \mathbf { b }$
$\mathbf { a } \times ( \mathbf { a } - \mathbf { b } )$

AQA Further Paper 2 2020 June Q2

2 Given that arg $( a + b \mathrm { i } ) = \varphi$, where $a$ and $b$ are positive real numbers and $0 < \varphi < \frac { \pi } { 2 }$, three of the following four statements are correct. Which statement is not correct? Tick $( \checkmark )$ one box. $$\begin{aligned} & \arg ( - a - b \mathrm { i } ) = \pi - \varphi
& \arg ( a - b \mathrm { i } ) = - \varphi
& \arg ( b + a \mathrm { i } ) = \frac { \pi } { 2 } - \varphi
& \arg ( b - a \mathrm { i } ) = \varphi - \frac { \pi } { 2 } \end{aligned}$$

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