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AQA Further Paper 1 2020 June Q8

6 marks Standard +0.8

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

13 marks Challenging +1.2

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}
1. Fird $\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}$

AQA Further Paper 1 2020 June Q10

10 marks Challenging +1.2

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 $y = \mathrm { f } ( x )$

AQA Further Paper 1 2020 June Q11

11 marks Standard +0.8

11 The lines $l _ { 1 } , l _ { 2 }$ and $l _ { 3 }$ are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11

Explain how you know that two of the lines are parallel.

(ii)

Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.

[5 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

Show that the lines $l _ { 1 }$ and $l _ { 3 }$ meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}

AQA Further Paper 1 2020 June Q12

8 marks Standard +0.8

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

12 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

6 marks 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}

Show that $$( r + 1 ) ^ { 2 } - r ^ { 2 } = 2 r + 1$$ 4
Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } ( 2 r + 1 ) = n ^ { 2 } + 2 n$$ 4
Verify that using the formula for $\sum _ { r = 1 } ^ { n } r$ gives the same result as that given in part (b).
[0pt] [3 marks]

AQA Further Paper 2 2021 June Q5

5 marks Standard +0.3

5 The equation $$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$ has roots $\alpha , \beta$ and $\gamma$ Find a cubic equation with roots $$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$

AQA Further Paper 2 2021 June Q6

8 marks Challenging +1.2

6 The ellipse $E _ { 1 }$ has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ $E _ { 1 }$ is translated by the vector $\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$ to give the ellipse $E _ { 2 }$ 6

Write down the equation of $E _ { 2 }$ 6
The ellipse $E _ { 3 }$ has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps $E _ { 2 }$ to $E _ { 3 }$ 6
Each of the lines $L _ { A }$ and $L _ { B }$ is a tangent to both $E _ { 2 }$ and $E _ { 3 }$ $L _ { A }$ is closer to the origin than $L _ { B }$ $E _ { 2 }$ and $E _ { 3 }$ both lie between $L _ { A }$ and $L _ { B }$ Sketch and label $E _ { 2 } , E _ { 3 } , L _ { A }$ and $L _ { B }$ on the axes below.
You do not need to show the values of the axis intercepts for $L _ { A }$ and $L _ { B }$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
Explain, without doing any calculations, why $L _ { A }$ has an equation of the form $$x + y = c$$ where $c$ is a constant.

AQA Further Paper 2 2021 June Q7

7 marks Challenging +1.8

7 \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-10_854_1027_264_520} The diagram shows a curve known as an astroid.
The curve has parametric equations $$\begin{aligned} & x = 4 \cos ^ { 3 } t \\ & y = 4 \sin ^ { 3 } t \\ & ( 0 \leq t < 2 \pi ) \end{aligned}$$ The section of the curve from $t = 0$ to $t = \frac { \pi } { 2 }$ is rotated through $2 \pi$ radians about the $x$-axis. Show that the curved surface area of the shape formed is equal to $\frac { b \pi } { c }$, where $b$ and $c$ are integers.

AQA Further Paper 2 2021 June Q8

6 marks Challenging +1.8

8 The complex number $z$ satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where $a$ is real.
Show that $a$ must lie in the interval $[ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]$, where $s$ and $t$ are prime numbers.
[0pt] [6 marks]

AQA Further Paper 2 2021 June Q9

14 marks Challenging +1.8

The line $L$ has polar equation $$r = \frac { 7 } { 4 } \sec \theta \quad \left( - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } \right)$$ Show that $L$ is perpendicular to the initial line.
9
The curve $C$ has polar equation $$r = 3 + \cos \theta \quad ( - \pi < \theta \leq \pi )$$ Find the polar coordinates of the points of intersection of $L$ and $C$ Fully justify your answer.
9
The region $R$ is the set of points such that
and $$r > \frac { 7 } { 4 } \sec \theta$$ Find the exact area of $R$ $$r < 3 + \cos \theta$$ Find the exact area of $R$ [0pt] [7 marks]

AQA Further Paper 2 2021 June Q10

13 marks Standard +0.3

10 In a colony of seabirds, there are $y$ birds at time $t$ years. 10

The rate of reduction in the number of birds due to birds dying or leaving the colony is proportional to the number of birds. In one year the reduction in the number of birds due to birds dying or leaving the colony is equal to $16 \%$ of the number of birds at the start of the year. If no birds are born or join the colony, find the constant $k$ such that $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k y$$ Give your answer to three significant figures.
10
A wildlife protection group takes measures to support the colony.
The rate of reduction in the number of birds due to birds dying or leaving the colony is the same as in part (a), but in addition:
Write down a first-order differential equation for $y$ and $t$ 10
The initial number of birds is 340 Solve your differential equation from part (b) to find $y$ in terms of $t$ 10
Describe two limitations of the model you have used. Limitation 1 $\_\_\_\_$ Limitation 2 $\_\_\_\_$

AQA Further Paper 2 2021 June Q11

9 marks Hard +2.3

11 The Cartesian equation of the line $L _ { 1 }$ is $$\frac { x + 1 } { 3 } = \frac { - y + 5 } { 2 } = \frac { 2 z + 5 } { 3 }$$ The Cartesian equation of the line $L _ { 2 }$ is $$\frac { 2 x - 1 } { 2 } = \frac { y - 14 } { m } = \frac { z + 12 } { p }$$ The non-singular matrix $\mathbf { N } = \left[ \begin{array} { c c c } - 0.5 & 1 & 2 \\ 1 & b & 4 \\ - 3 & - 2 & c \end{array} \right]$ maps the line $L _ { 1 }$ onto the line $L _ { 2 }$ Calculate the values of the constants $b , c , m$ and $p$ Fully justify your answers.

AQA Further Paper 2 2021 June Q12

12 marks Challenging +1.2

12 The integral $S _ { n }$ is defined by $$S _ { n } = \int _ { 0 } ^ { a } x ^ { n } \sinh x \mathrm {~d} x \quad ( n \geq 0 )$$ 12

Show that for $n \geq 2$ $$S _ { n } = n ( n - 1 ) S _ { n - 2 } + a ^ { n } \cosh a - n a ^ { n - 1 } \sinh a$$
12
Hence show that $\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24$

AQA Further Paper 2 2021 June Q13

16 marks Challenging +1.8

Two of the solutions to the equation $\cos 6 \theta = 0$ are $\theta = \frac { \pi } { 4 }$ and $\theta = \frac { 3 \pi } { 4 }$ Find the other solutions to the equation $\cos 6 \theta = 0$ for $0 \leq \theta \leq \pi$ 13
Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
Use the fact that $\theta = \frac { \pi } { 4 }$ and $\theta = \frac { 3 \pi } { 4 }$ are solutions to the equation $\cos 6 \theta = 0$ to find a factor of $32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$ in the form ( $a \cos ^ { 2 } \theta + b$ ), where $a$ and $b$ are integers.
[0pt] [4 marks]
Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}