AQA Further AS Paper 1 (Further AS Paper 1) 2021 June

1 The complex number \(\omega\) is shown below on the Argand diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-02_597_650_632_689} Which of the following complex numbers could be \(\omega\) ?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \cos ( - 2 ) + i \sin ( - 2 )
& \cos ( - 1 ) + i \sin ( - 1 )
& \cos ( 1 ) + i \sin ( 1 )
& \cos ( 2 ) + i \sin ( 2 ) \end{aligned}$$ □
□
□
□

2 Given that \(\mathrm { f } ( x ) = 3 x - 1\) find the mean value of \(\mathrm { f } ( x )\) over the interval \(4 \leq x \leq 8\) Circle your answer. 6111717

3 The matrix \(\mathbf { M }\) represents a rotation about the \(x\)-axis. $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 0 & 0
0 & a & \frac { \sqrt { 3 } } { 2 }
0 & b & - \frac { 1 } { 2 } \end{array} \right]$$ Which of the following pairs of values is correct?
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } a = \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square
a = \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \square
a = - \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square
a = - \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \end{array}$$

4 The point \(( 2 , - 1 )\) is invariant under the transformation represented by the matrix \(\mathbf { N }\) Which of the following matrices could be \(\mathbf { N }\) ? Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { l l } 4 & 6
2 & 5 \end{array} \right]\)
\(\left[ \begin{array} { l l } 6 & 5
4 & 2 \end{array} \right]\)
\(\left[ \begin{array} { l l } 5 & 2
6 & 4 \end{array} \right]\)
\(\left[ \begin{array} { l l } 2 & 4
5 & 6 \end{array} \right]\)

5 Show that the vectors \(\left[ \begin{array} { c } 1
- 3
5 \end{array} \right]\) and \(\left[ \begin{array} { l } 7
4
1 \end{array} \right]\) are perpendicular.

6 Prove the identity $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$

7 Show that the Maclaurin series for \(\ln ( \mathrm { e } + 2 \mathrm { e } x )\) is $$1 + 2 x - 2 x ^ { 2 } + a x ^ { 3 } - \ldots$$ where \(a\) is to be determined.

8 Stephen is correctly told that \(( 1 + \mathrm { i } )\) and - 1 are two roots of the polynomial equation $$z ^ { 3 } - 2 \mathrm { i } z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are complex numbers.
8

1. Stephen states that ( \(1 - \mathrm { i }\) ) must also be a root of the equation because roots of polynomial equations occur in conjugate pairs. Explain why Stephen's reasoning is wrong. 8
2. \(\quad\) Find \(p\) and \(q\)

9

1. Use the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = a n ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are constants to be determined.
   [0pt] [4 marks]
   9
2. Hence, or otherwise, find a fully factorised expression for $$\sum _ { r = n + 1 } ^ { 5 n } r ( r + 3 )$$ $$\mathbf { A } = \left[ \begin{array} { c c } 3 & i - 1
   i & 2 \end{array} \right]$$

10

1. Show that \(\operatorname { det } \mathbf { A } = a + \mathrm { i }\) where \(a\) is an integer to be determined. 10 Matrix A is given by 10
2. Matrix B is given by $$\mathbf { B } = \left[ \begin{array} { c c } 14 - 2 \mathrm { i } & b
   c & d \end{array} \right] \quad \text { and } \quad \mathbf { A B } = p$$ where \(b , c , d \in \mathbb { C }\) and \(p \in \mathbb { N }\)
   Find \(b , c , d\) and \(p\)

11

1. Show that, for all positive integers \(r\), $$\frac { 1 } { ( r - 1 ) ! } - \frac { 1 } { r ! } = \frac { r - 1 } { r ! }$$ ⟶
   11
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { r - 1 } { r ! } = a + \frac { b } { n ! }$$ where \(a\) and \(b\) are integers to be determined.

12 The equation \(x ^ { 3 } - 2 x ^ { 2 } - x + 2 = 0\) has three roots. One of the roots is 2 12

1. Find the other two roots of the equation. 12
2. Hence, or otherwise, solve $$\cosh ^ { 3 } \theta - 2 \cosh ^ { 2 } \theta - \cosh \theta + 2 = 0$$ giving your answers in an exact form.

13 Prove by induction that, for all integers \(n \geq 1\) $$\sum _ { r = 1 } ^ { n } 2 ^ { - r } = 1 - 2 ^ { - n }$$ [4 marks]

14 Curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$ 14

1. Curve \(C _ { 2 }\) is a reflection of \(C _ { 1 }\) in the line \(y = x\)
   Write down an equation of \(C _ { 2 }\)
   14
2. Curve \(C _ { 3 }\) is a circle of radius 4 , centred at the origin.
   Describe a single transformation which maps \(C _ { 1 }\) onto \(C _ { 3 }\)
   14
3. Curve \(C _ { 4 }\) is a translation of \(C _ { 1 }\)
   The positive \(x\)-axis and the positive \(y\)-axis are tangents to \(C _ { 4 }\)
   14
4. 1. Sketch the graphs of \(C _ { 1 }\) and \(C _ { 4 }\) on the axes opposite. Indicate the coordinates of the \(x\) and \(y\) intercepts on your graphs.
      [0pt] [2 marks]
      14
5. (ii) Determine the translation vector.
   [0pt] [2 marks]
   14
6. (iii) The line \(y = m x + c\) is a tangent to both \(C _ { 1 }\) and \(C _ { 4 }\) Find the value of \(m\)

15 Two submarines are travelling on different straight lines. The two lines are described by the equations $$\mathbf { r } = \left[ \begin{array} { c } 2
- 1
4 \end{array} \right] + \lambda \left[ \begin{array} { c } 5
3
- 2 \end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$ 15

1. 1. Show that the two lines intersect.
      [0pt] [3 marks]
      15
2. (ii) Find the position vector of the point of intersection.
   15
3. Tracey says that the submarines will collide because there is a common point on the two lines. Explain why Tracey is not necessarily correct. 15
4. Calculate the acute angle between the lines $$\mathbf { r } = \left[ \begin{array} { c } 2
   - 1
   4 \end{array} \right] + \lambda \left[ \begin{array} { c } 5
   3
   - 2 \end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$ Give your angle to the nearest \(0.1 ^ { \circ }\)

16 Curve \(C\) has equation \(y = \frac { a x } { x + b }\) where \(a\) and \(b\) are constants.
The equations of the asymptotes to \(C\) are \(x = - 2\) and \(y = 3\)
\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-20_796_819_459_609} 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The gradient of \(C\) at the origin is \(\frac { 3 } { 2 }\)
   With reference to the graph, explain why there is exactly one root of the equation $$\frac { a x } { x + b } = \frac { 3 x } { 2 }$$ 16
3. Using the values found in part (a), solve the inequality $$\frac { a x } { x + b } \leq 1 - x$$ [4 marks]

17 The curve \(C _ { 1 }\) has polar equation \(r = 2 a ( 1 + \sin \theta )\) for \(- \pi < \theta \leq \pi\) where \(a\) is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-22_469_830_402_605} The point \(M\) lies on \(C _ { 1 }\) and the initial line.
17

1. Write down, in terms of \(a\), the polar coordinates of \(M\) 17
2. \(\quad N\) is the point on \(C _ { 1 }\) that is furthest from the pole \(O\)
   Find, in terms of \(a\), the polar coordinates of \(N\)
   17
3. The curve \(C _ { 2 }\) has polar equation \(r = 3 a\) for \(- \pi < \theta \leq \pi\) \(C _ { 2 }\) intersects \(C _ { 1 }\) at points \(P\) and \(Q\) Show that the area of triangle \(N P Q\) can be written in the form $$m \sqrt { 3 } a ^ { 2 }$$ where \(m\) is a rational number to be determined.
   17
4. On the initial line below, sketch the graph of \(r = 2 a ( 1 + \cos \theta )\) for \(- \pi < \theta \leq \pi\) Include the polar coordinates, in terms of \(a\), of any intersection points with the initial line.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-24_65_657_1425_991}
   \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-25_2492_1721_217_150}