AQA Further AS Paper 1 (Further AS Paper 1) 2020 June

1 Express the complex number \(1 - \mathrm { i } \sqrt { 3 }\) in modulus-argument form.
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) & \square
2 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) & \square
2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) & \square
2 \left( \cos \left( - \frac { 2 \pi } { 3 } \right) + i \sin \left( - \frac { 2 \pi } { 3 } \right) \right) \end{array}$$

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. $$\begin{aligned} & - 1 + i \text { and } - 1
& 1 + i \text { and } 1
& - 1 + i \text { and } 1
& 1 + i \text { and } - 1 \end{aligned}$$ □
□
□
□

3 Given \(( x - 1 ) ( x - 2 ) ( x - a ) < 0\) and \(a > 2\) Find the set of possible values of \(x\).
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \{ x : x < 1 \} \cup \{ x : 2 < x < a \}
& \{ x : 1 < x < 2 \} \cup \{ x : x > a \}
& \{ x : x < - a \} \cup \{ x : - 2 < x < - 1 \}
& \{ x : - a < x < - 2 \} \cup \{ x : x > - 1 \} \end{aligned}$$

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 \(a = - 1\)

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 $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$

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

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.

9 The quadratic equation \(2 x ^ { 2 } + p x + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\). 9

1. 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 \(\beta + 1\)

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}

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}

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 \(4 \leq x \leq 8\)

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}

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 $$\frac { x + 7 } { x + 1 } \leq x + 1$$

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.

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.

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]

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}