AQA Further AS Paper 1 (Further AS Paper 1) 2024 June

1 Express \(\cosh ^ { 2 } x\) in terms of \(\sinh x\)
Circle your answer.
\(1 + \sinh ^ { 2 } x\)
\(1 - \sinh ^ { 2 } x\)
\(\sinh ^ { 2 } x - 1\)
\(- 1 - \sinh ^ { 2 } x\)

2 The function f is defined by $$f ( x ) = 2 x + 3 \quad 0 \leq x \leq 5$$ The region \(R\) is enclosed by \(y = \mathrm { f } ( x ) , x = 5\), the \(x\)-axis and the \(y\)-axis.
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Give an expression for the volume of the solid formed.
Tick ( ✓ ) one box.
\(\pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) d x\)
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-02_113_108_1539_1000}
\(\pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) ^ { 2 } \mathrm {~d} x\)
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-02_115_108_1699_1000}
\(2 \pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) d x\) □
\(2 \pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) ^ { 2 } \mathrm {~d} x\) □

3 The matrix \(\mathbf { A }\) is such that \(\operatorname { det } ( \mathbf { A } ) = 2\) Determine the value of \(\operatorname { det } \left( \mathbf { A } ^ { - 1 } \right)\)
Circle your answer.
-2
\(- \frac { 1 } { 2 }\)
\(\frac { 1 } { 2 }\)
2

4 The line \(L\) has vector equation $$\mathbf { r } = \left[ \begin{array} { c } 4
- 7
0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 9
1
3 \end{array} \right]$$ Give the equation of \(L\) in Cartesian form.
Tick ( ✓ ) one box.
\(\frac { x + 4 } { - 9 } = \frac { y - 7 } { 1 } = \frac { z } { 3 }\)
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-03_108_109_1398_993}
\(\frac { x - 4 } { - 9 } = \frac { y + 7 } { 1 } = \frac { z } { 3 }\)
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-03_108_111_1567_991}
\(\frac { x + 9 } { 4 } = \frac { y - 1 } { - 7 } , z = 3\) □
\(\frac { x - 9 } { 4 } = \frac { y + 1 } { - 7 } , z = 3\) □

5 The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by $$\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \mathbf { b } = 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }$$ 5

1. Calculate a.b 5
2. \(\quad\) Calculate \(| \mathbf { a } |\) and \(| \mathbf { b } |\)
   \(| \mathbf { a } | =\) \(\_\_\_\_\)
   5
3. Calculate the acute angle between \(\mathbf { a }\) and \(\mathbf { b }\)
   Give your answer to the nearest degree.

6

1. On the axes below, sketch the graph of $$y = \cosh x$$ Indicate the value of any intercept of the curve with the axes.
   \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-05_1114_1121_552_447} 6
2. Solve the equation $$\cosh x = 2$$ Give your answers to three significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-06_2491_1755_173_123}

7 The function f is defined by $$f ( x ) = \frac { 1 } { \sqrt { x } } \quad 4 \leq x \leq 7$$ Find the mean value of f over the interval \(4 \leq x \leq 7\) Give your answer in exact form.

8

1. The complex number \(z\) is given by \(z = x + i y\) where \(x , y \in \mathbb { R }\) 8
2. 1. Write down the complex conjugate \(z ^ { * }\) in terms of \(x\) and \(y\) 8
3. (ii) Hence prove that \(z z ^ { * }\) is real for all \(z \in \mathbb { C }\)
   8
4. The complex number \(w\) satisfies the equation $$3 w + 10 \mathrm { i } = 2 w ^ { \star } + 5$$ 8
5. 1. Find \(w\)
      8
6. (ii) Calculate the value of \(w ^ { 2 } \left( w ^ { * } \right) ^ { 2 }\)

9

1. Show that, for all positive integers \(r\), $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) } = \frac { n } { a n + b }$$ where \(a\) and \(b\) are integers to be determined.
   9
3. Hence find the exact value of $$\sum _ { r = 1001 } ^ { 2000 } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$ \(\_\_\_\_\) The curve \(C\) has equation $$y = \frac { 2 x - 10 } { 3 x - 5 }$$ Figure 1 shows the curve \(C\) with its asymptotes. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-12_979_1079_641_468}
   \end{figure}

10

1. Write down the equations of the asymptotes of \(C\)
   10
2. The line \(L\) has equation $$y = - \frac { 2 } { 5 } x + 2$$ 10
3. 1. Draw the line \(L\) on Figure 1 10
4. (ii) Hence, or otherwise, solve the inequality $$\frac { 2 x - 10 } { 3 x - 5 } \leq - \frac { 2 } { 5 } x + 2$$

11 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } 3 \mathrm { i } & - 2
a & - \mathrm { i } \end{array} \right] \quad \text { and } \quad \mathbf { B } = \left[ \begin{array} { c c } 4 & 5
- 2 \mathrm { i } & - 1 \end{array} \right]$$ where \(a\) is a real number. Calculate the product \(\mathbf { A B }\) in terms of \(a\)
Give your answer in its simplest form.
[0pt] [3 marks]

12 Prove by induction that, for all \(n \in \mathbb { N }\), the expression $$5 ^ { n } - 2 ^ { n }$$ is divisible by 3
[0pt] [4 marks]
LL

13 The cubic equation \(x ^ { 3 } - x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\) The cubic equation \(\mathrm { p } ( x ) = 0\) has roots \(\alpha - 1 , \beta - 1\) and \(\gamma - 1\)
The coefficient of \(x ^ { 3 }\) in \(\mathrm { p } ( x )\) is 1 13

1. Describe fully the transformation that maps the graph of \(y = x ^ { 3 } - x - 7\) onto the graph of \(y = \mathrm { p } ( x )\)
   13
2. Find \(\mathrm { p } ( x )\)
   Turn over for the next question

14 The matrix \(\mathbf { M }\) represents the transformation T , and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 & - 1
- 2 & 6 \end{array} \right]$$ 14

1. The point \(A\) has coordinates ( \(4 , - 5\) )
   Find the coordinates of the image of \(A\) under T
   14
2. Show that the only invariant point under T is the origin.
   14
3. The line \(L _ { 1 }\) has equation \(y = x + 1\) The transformation \(T\) maps the line \(L _ { 1 }\) onto the line \(L _ { 2 }\)
   Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\)

15

1. Use Maclaurin's series expansion for \(\ln ( 1 + x )\) to show that the first three terms of the Maclaurin's series expansion of \(\ln ( 1 + 3 x )\) are $$3 x - \frac { 9 } { 2 } x ^ { 2 } + 9 x ^ { 3 }$$ 15
2. Julia attempts to use the series expansion found in part (a) to find an approximation for \(\ln 4\) Julia's incorrect working is shown below. $$\begin{array} { r } \text { Let } 1 + 3 x = 4
   3 x = 3
   x = 1 \end{array}$$ $$\text { So } \begin{aligned} \ln 4 & \approx 3 \times 1 - \frac { 9 } { 2 } \times 1 ^ { 2 } + 9 \times 1 ^ { 3 }
   & \approx 3 - 4.5 + 9
   & \approx 7.5 \end{aligned}$$ Explain the error in Julia's working.
   15
3. Use \(x = - \frac { 1 } { 6 }\) in the series expansion found in part (a) to find an approximation for \(\ln 4\) Fully justify your answer.

16 The curve \(C\) has the polar equation $$r = \frac { 2 } { \sqrt { \cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta } } \quad - \pi < \theta \leq \pi$$ 16

1. Show that the Cartesian equation of \(C\) can be written as $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive integers to be determined.
   [0pt] [4 marks]
   16
2. Hence sketch the graph of \(C\) on the axes below. Indicate the value of any intercepts of the curve with the axes.
   \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-23_1122_1121_452_447}

17 The circle \(C\) represents the locus of points satisfying the equation $$| z - a \mathrm { i } | = b$$ where \(a\) and \(b\) are real constants. The circle \(C\) intersects the imaginary axis at 2 i and 8 i
The circle \(C\) is shown on the Argand diagram in Figure 2 \begin{figure}[h] \end{figure} 17

1. 1. Write down the value of \(a\) 17
2. (ii) Write down the value of \(b\)
   17
3. The half-line \(L\) represents the locus of points satisfying the equation $$\arg ( z ) = \tan ^ { - 1 } ( k )$$ where \(k\) is a positive constant.
   The point \(P\) is the only point which lies on both \(C\) and \(L\), as shown in Figure 3 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-25_766_770_685_699}
   \end{figure} 17
4. 1. The point \(O\) represents the number \(0 + 0 \mathrm { i }\)
      Calculate the length \(O P\)
      17
5. (ii) Calculate the exact value of \(k\)
   17
6. (iii) Find the complex number represented by point \(P\)
   Give your answer in the form \(x + y i\) where \(x\) and \(y\) are real.