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AQA Further AS Paper 1 2019 June Q7
5 marks Standard +0.3
7
  1. Show that $$\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { A } { r ^ { 2 } - 1 }$$ where \(A\) is a constant to be found. 7
  2. Hence use the method of differences to show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } \equiv \frac { a n ^ { 2 } + b n + c } { 4 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be found.
AQA Further AS Paper 1 2019 June Q8
7 marks
8 Given that \(z _ { 1 } = 2 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)\) and \(z _ { 2 } = 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)\) 8
  1. Find the value of \(\left| z _ { 1 } z _ { 2 } \right|\) 8
  2. Find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) 8
  3. Sketch \(z _ { 1 }\) and \(z _ { 2 }\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively.
    [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-10_764_869_1546_587} 8
  4. A third complex number \(w\) satisfies both \(| w | = 2\) and \(- \pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(P \widehat { R } Q\). Fully justify your answer.
AQA Further AS Paper 1 2019 June Q9
7 marks Standard +0.8
9
  1. Saul is solving the equation $$2 \cosh x + \sinh ^ { 2 } x = 1$$ He writes his steps as follows: $$\begin{aligned} 2 \cosh x + \sinh ^ { 2 } x & = 1 \\ 2 \cosh x + 1 - \cosh ^ { 2 } x & = 1 \\ 2 \cosh x - \cosh ^ { 2 } x & = 0 \\ \cosh x \neq 0 \therefore 2 - \cosh x & = 0 \\ \cosh x & = 2 \\ x & = \pm \cosh ^ { - 1 } ( 2 ) \end{aligned}$$ Identify and explain the error in Saul's method. 9
  2. Anna is solving the different equation
    g (b) Anna is solving the different equation $$\sinh ^ { 2 } ( 2 x ) - 2 \cosh ( 2 x ) = 1$$ and finds the correct answers in the form \(x = \frac { 1 } { p } \cosh ^ { - 1 } ( q + \sqrt { r } )\), where \(p , q\) and \(r\) are integers. Find the possible values of \(p , q\) and \(r\).
    Fully justify your answer.
AQA Further AS Paper 1 2019 June Q10
6 marks Standard +0.3
10
  1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(\mathrm { e } ^ { x }\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). 10
  2. Hence find a trigonometric function for which the first three terms of its Maclaurin series are the same as the first three terms of the Maclaurin series for cosh (ix).
    [0pt] [3 marks] \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-15_2488_1716_219_153}
AQA Further AS Paper 1 2019 June Q11
8 marks Moderate -0.8
11
  1. Curve \(C\) has equation $$y = \frac { x ^ { 2 } + p x - q } { x ^ { 2 } - r }$$ where \(p , q\) and \(r\) are positive constants.
    Write down the equations of its asymptotes.
AQA Further AS Paper 1 2019 June Q12
12 marks Standard +0.3
12 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$$ 12
  1. Prove by induction that, for all integers \(n \geq 1\), $$\mathbf { A } ^ { n } = \left[ \begin{array} { c c } 1 & 3 ^ { n } - 1 \\ 0 & 3 ^ { n } \end{array} \right]$$ 12
  2. Find all invariant lines under the transformation matrix \(A\). Fully justify your answer.
    12
  3. Find a line of invariant points under the transformation matrix \(\mathbf { A }\).
AQA Further AS Paper 1 2019 June Q13
10 marks Standard +0.3
13 Line \(l _ { 1 }\) has Cartesian equation $$x - 3 = \frac { 2 y + 2 } { 3 } = 2 - z$$ 13
  1. Write the equation of line \(l _ { 1 }\) in the form $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$$ where \(\lambda\) is a parameter and \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.
    13
  2. Line \(l _ { 2 }\) passes through the points \(P ( 3,2,0 )\) and \(Q ( n , 5 , n )\), where \(n\) is a constant.
    13
    1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
      13
    2. (ii) Explain briefly why lines \(l _ { 1 }\) and \(l _ { 2 }\) cannot be parallel.
      3. 13
    3. (iii) Given that \(\theta\) is the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\), show that
      4. \(\cos \theta = \frac { p } { \sqrt { 34 n ^ { 2 } + q n + 306 } }\)
      where \(p\) and \(q\) are constants to be found.
AQA Further AS Paper 1 2019 June Q14
7 marks Standard +0.8
14 The graph of \(y = x ^ { 3 } - 3 x\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-22_718_771_370_632} The two stationary points have \(x\)-coordinates of - 1 and 1
The cubic equation $$x ^ { 3 } - 3 x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha , \beta\) and \(\gamma\).
The roots \(\alpha\) and \(\beta\) are not real.
14
  1. Explain why \(\alpha + \beta = - \gamma\) 14
  2. Find the set of possible values for the real constant \(p\).
    14
  3. \(\quad \mathrm { f } ( x ) = 0\) is a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\) 14
    1. Show that the constant term of \(\mathrm { f } ( x )\) is \(p + 2\) 14
  5. (ii) Write down the \(x\)-coordinates of the stationary points of \(y = \mathrm { f } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-24_2488_1719_219_150} Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further AS Paper 1 2020 June Q1
1 marks Easy -1.2
1 Express the complex number \(1 - \mathrm { i } \sqrt { 3 }\) in modulus-argument form.
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) & \square \\ 2 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) & \square \\ 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) & \square \\ 2 \left( \cos \left( - \frac { 2 \pi } { 3 } \right) + i \sin \left( - \frac { 2 \pi } { 3 } \right) \right) \end{array}$$
AQA Further AS Paper 1 2020 June Q2
1 marks Moderate -0.8
2 Given that \(1 - \mathrm { i }\) is a root of the equation \(z ^ { 3 } - 3 z ^ { 2 } + 4 z - 2 = 0\), find the other two roots. Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & - 1 + i \text { and } - 1 \\ & 1 + i \text { and } 1 \\ & - 1 + i \text { and } 1 \\ & 1 + i \text { and } - 1 \end{aligned}$$ □
□
□
□
AQA Further AS Paper 1 2020 June Q3
1 marks Moderate -0.8
3 Given \(( x - 1 ) ( x - 2 ) ( x - a ) < 0\) and \(a > 2\) Find the set of possible values of \(x\).
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \{ x : x < 1 \} \cup \{ x : 2 < x < a \} \\ & \{ x : 1 < x < 2 \} \cup \{ x : x > a \} \\ & \{ x : x < - a \} \cup \{ x : - 2 < x < - 1 \} \\ & \{ x : - a < x < - 2 \} \cup \{ x : x > - 1 \} \end{aligned}$$
AQA Further AS Paper 1 2020 June Q4
5 marks Moderate -0.5
4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that $$\mathbf { A } = \left[ \begin{array} { c c c } 2 & a & 3 \\ 0 & - 2 & 1 \end{array} \right] \quad \text { and } \quad \mathbf { B } = \left[ \begin{array} { c c } 1 & - 3 \\ - 2 & 4 a \\ 0 & 5 \end{array} \right]$$ 4
  1. Find the product \(\mathbf { A B }\) in terms of \(a\).
    [0pt] [2 marks]
    4
  2. Find the determinant of \(\mathbf { A B }\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-04_31_31_513_367}
    "
    □ \(\quad \mathbf { A } = \left[ \begin{array} { c c c } 2 & a & 3 \\ 0 & - 2 & 1 \end{array} \right]\) and \(\quad \mathbf { B } = \left[ \begin{array} { c c } 1 & - 3 \\ - 2 & 4 a \\ 0 & 5 \end{array} \right]\) \(\mathbf { 4 }\) (a) Find the product \(\mathbf { A B }\) in terms of \(a\). 4
  3. Show that \(\mathbf { A B }\) is singular when \(a = - 1\)
AQA Further AS Paper 1 2020 June Q5
4 marks Standard +0.8
5
  1. Show that $$r ^ { 2 } ( r + 1 ) ^ { 2 } - ( r - 1 ) ^ { 2 } r ^ { 2 } = p r ^ { 3 }$$ where \(p\) is an integer to be found.
    [0pt] [1 mark]
    5
  2. Hence use the method of differences to show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
AQA Further AS Paper 1 2020 June Q6
2 marks Moderate -0.5
6 Anna has been asked to describe the transformation given by the matrix $$\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$ She writes her answer as follows: The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where $$\begin{gathered} \sin \theta = \frac { 1 } { 2 } \quad \text { and } \quad - \sin \theta = - \frac { 1 } { 2 } \\ \theta = 30 ^ { \circ } \end{gathered}$$ Identify and correct the error in Anna's work.
[0pt] [2 marks] \(7 \quad\) Prove by induction that, for all integers \(n \geq 1\), the expression \(7 ^ { n } - 3 ^ { n }\) is divisible by 4
AQA Further AS Paper 1 2020 June Q8
8 marks Standard +0.3
8
  1. Prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\) 8
  2. Prove that the graphs of $$y = \sinh x \quad \text { and } \quad y = \cosh x$$ do not intersect.
AQA Further AS Paper 1 2020 June Q9
8 marks Standard +0.3
9 The quadratic equation \(2 x ^ { 2 } + p x + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\). 9
    1. Write down the value of \(\alpha \beta\). 9
  2. (ii) Express \(\alpha + \beta\) in terms of \(p\). 9
  3. Hence find \(( \alpha - \beta ) ^ { 2 }\) in terms of \(p\).
    9
  4. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\)
AQA Further AS Paper 1 2020 June Q10
8 marks Standard +0.3
10
  1. Show that the equation $$y = \frac { 3 x - 5 } { 2 x + 4 }$$ can be written in the form $$( x + a ) ( y + b ) = c$$ where \(a\), \(b\) and \(c\) are integers to be found.
    10
  2. Write down the equations of the asymptotes of the graph of $$y = \frac { 3 x - 5 } { 2 x + 4 }$$ 10
  3. Sketch, on the axes provided, the graph of $$y = \frac { 3 x - 5 } { 2 x + 4 }$$
    \includegraphics[max width=\textwidth, alt={}]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-15_1104_1115_1439_466}
AQA Further AS Paper 1 2020 June Q11
3 marks Challenging +1.2
11 Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2 \pi\) \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-16_81_821_1854_918}
AQA Further AS Paper 1 2020 June Q12
2 marks Standard +0.8
12 The mean value of the function f over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathrm { g } ( x )\) is a reflection in the \(x\)-axis of \(y = \mathrm { f } ( x )\).
The graph of \(y = \mathrm { h } ( x )\) is a translation of \(y = \mathrm { g } ( x )\) by \(\left[ \begin{array} { l } 3 \\ 7 \end{array} \right]\) Determine, in terms of \(m\), the mean value of the function h over the interval \(4 \leq x \leq 8\)
AQA Further AS Paper 1 2020 June Q13
9 marks Standard +0.3
13 Line \(l _ { 1 }\) has equation $$\frac { x - 2 } { 3 } = \frac { 1 - 2 y } { 4 } = - z$$ and line \(l _ { 2 }\) has equation $$\mathbf { r } = \left[ \begin{array} { c } - 7 \\ 4 \\ - 2 \end{array} \right] + \mu \left[ \begin{array} { c } 12 \\ a + 3 \\ 2 b \end{array} \right]$$ 13
  1. In the case when \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, show that \(a = - 11\) and find the value of \(b\).
    \includegraphics[max width=\textwidth, alt={}]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-19_2484_1712_219_150}
AQA Further AS Paper 1 2020 June Q14
7 marks Standard +0.3
14
  1. Given $$\frac { x + 7 } { x + 1 } \leq x + 1$$ show that $$\frac { ( x + a ) ( x + b ) } { x + c } \geq 0$$ where \(a , b\), and \(c\) are integers to be found.
    14
  2. Briefly explain why this statement is incorrect. $$\frac { ( x + p ) ( x + q ) } { x + r } \geq 0 \Leftrightarrow ( x + p ) ( x + q ) ( x + r ) \geq 0$$ 14
  3. Solve $$\frac { x + 7 } { x + 1 } \leq x + 1$$
AQA Further AS Paper 1 2020 June Q15
4 marks Moderate -0.5
15 A segment of the line \(y = k x\) is rotated about the \(x\)-axis to generate a cone with vertex \(O\). The distance of \(O\) from the centre of the base of the cone is \(h\).
The radius of the base of the cone is \(r\). \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-22_629_1006_566_516} 15
  1. Find \(k\) in terms of \(r\) and \(h\).
    15
  2. Use calculus to prove that the volume of the cone is $$\frac { 1 } { 3 } \pi r ^ { 2 } h$$ \(16 \quad \mathbf { A }\) and \(\mathbf { B }\) are non-singular square matrices.
AQA Further AS Paper 1 2020 June Q16
4 marks Easy -1.2
16
  1. Write down the product \(\mathbf { A A } ^ { - 1 }\) as a single matrix.
    [0pt] [1 mark]
    16
  2. \(\quad \mathbf { M }\) is a matrix such that \(\mathbf { M } = \mathbf { A B }\).
    Prove that \(\mathbf { M } ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) [0pt] [3 marks]
    The polar equation of the circle \(C\) is Find, in terms of \(a\), the radius of \(C\). Fully justify your answer.
AQA Further AS Paper 1 2020 June Q17
4 marks Standard +0.8
17 The polar equation of the circle \(C\) is $$r = a ( \cos \theta + \sin \theta )$$ Find, in terms of \(a\), the radius of \(C\).
Fully justify your answer. \(\_\_\_\_\) [4 marks]
AQA Further AS Paper 1 2020 June Q18
5 marks Challenging +1.2
18 The locus of points \(L _ { 1 }\) satisfies the equation \(| z | = 2\) The locus of points \(L _ { 2 }\) satisfies the equation \(\arg ( z + 4 ) = \frac { \pi } { 4 }\) 18
  1. Sketch \(L _ { 1 }\) on the Argand diagram below.
    [0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-26_1152_1195_644_427} 18
  2. Sketch \(L _ { 2 }\) on the Argand diagram above.
    [0pt] [1 mark] 18
  3. The complex number \(a + \mathrm { i } b\), where \(a\) and \(b\) are real, lies on \(L _ { 1 }\) The complex number \(c + \mathrm { i } d\), where \(c\) and \(d\) are real, lies on \(L _ { 2 }\) Calculate the least possible value of the expression $$( c - a ) ^ { 2 } + ( d - b ) ^ { 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-28_2492_1721_217_150}