AQA Further AS Paper 1 (Further AS Paper 1) 2022 June

1 Which of the following exponential expressions is equivalent to \(2 \sinh x\) ?
Circle your answer. \(\mathrm { e } ^ { x }\) \(\mathrm { e } ^ { x } + \mathrm { e } ^ { - x }\) \(\mathrm { e } ^ { x } - \mathrm { e } ^ { - x }\) \(\mathrm { e } ^ { - x }\)

2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\) Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
[0pt] [1 mark] \(p - p - q - q\)

3 Which of the following transformations is represented by the matrix \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) ?
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] Rotation of \(180 ^ { \circ }\) about the \(x\)-axis □ Reflection in the plane \(x = 0\) □ Rotation of \(180 ^ { \circ }\) about the \(y\)-axis □ Reflection in the plane \(y = 0\) □

4 The complex numbers \(w\) and \(z\) are defined as $$\begin{aligned} w & = 2 ( \cos \alpha + \mathrm { i } \sin \alpha ) \\ z & = 3 ( \cos \beta + \mathrm { i } \sin \beta ) \end{aligned}$$ Find the product \(w z\) Tick \(( \checkmark )\) one box. $$\begin{aligned} & 5 ( \cos ( \alpha \beta ) + \mathrm { i } \sin ( \alpha \beta ) ) \\ & 6 ( \cos ( \alpha \beta ) + \mathrm { i } \sin ( \alpha \beta ) ) \\ & 5 ( \cos ( \alpha + \beta ) + \mathrm { i } \sin ( \alpha + \beta ) ) \\ & 6 ( \cos ( \alpha + \beta ) + \mathrm { i } \sin ( \alpha + \beta ) ) \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_113_113_762_1206} \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_108_108_900_1206} \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_113_113_1032_1206}
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5 Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) is \(2 + 11 \mathrm { i }\) [0pt] [3 marks]

6 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left[ \begin{array} { c c } 5 & 2 \\ - 3 & 4 \end{array} \right]$$ 6

1. \(\quad\) Find \(\operatorname { det } \mathbf { A }\) 6
2. Find \(\mathbf { A } ^ { - 1 }\) 6
3. Given that \(\mathbf { A B } = \left[ \begin{array} { c c } 9 & 6 \\ 5 & 12 \end{array} \right]\) and \(\mathbf { M } = 2 \mathbf { A } + \mathbf { B }\) find the matrix \(\mathbf { M }\)

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left[ \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right] \\ & l _ { 2 } : \mathbf { r } = \left[ \begin{array} { c } - 12 \\ a \\ - 3 \end{array} \right] + \mu \left[ \begin{array} { c } 3 \\ 2 \\ - 1 \end{array} \right] \end{aligned}$$ 7

1. Show that the point \(P ( - 3,9 , - 4 )\) lies on \(l _ { 1 }\) 7
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\) 7
3. Given that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, calculate the value of the constant \(a\) 7
4. Hence, find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\)

8 The curve \(C\) has the polar equation $$r = 4 - 2 \cos \theta \quad - \pi < \theta \leq \pi$$ 8

1. Verify that the point with polar coordinates \(\left( 3 , \frac { \pi } { 3 } \right)\) lies on \(C\) 8
2. Find the exact polar coordinates of the point on \(C\) which is furthest from the pole, \(O\) [3 marks]
   8
3. Find the exact Cartesian coordinates of the point on \(C\) where \(\theta\) is \(\frac { \pi } { 6 }\)

9

1. Show that, for \(r > 0\), $$\ln ( r + 2 ) - \ln r = \ln \left( 1 + \frac { 2 } { r } \right)$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \ln \left( 1 + \frac { 2 } { r } \right) = \ln \left( \frac { 1 } { 2 } ( n + a ) ( n + b ) \right)$$ where \(a\) and \(b\) are integers to be found.

10 The diagram below shows an ellipse \(E\) The coordinate axes are the lines of symmetry of \(E\) \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-14_645_780_450_630} 10

1. Write down an equation of \(E\) 10
2. The region bounded by the \(x\)-axis and the ellipse \(E\) for \(y \geq 0\) is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-15_643_775_408_635} A solid \(S\) is formed by rotating the shaded region through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of \(S\) is \(a \pi\) where \(a\) is an integer to be found. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-16_2488_1732_219_139}

11 Prove by induction that, for all integers \(n \geq 1\), $$\left( \mathbf { A B A } ^ { - 1 } \right) ^ { n } = \mathbf { A B } ^ { n } \mathbf { A } ^ { - 1 }$$ where \(\mathbf { A }\) and \(\mathbf { B }\) are square matrices of equal dimensions, and \(\mathbf { A }\) is non-singular.

12

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 2 \mathrm { i } | = 2$$
   \includegraphics[max width=\textwidth, alt={}]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-18_1219_1260_477_392}
   12
2. Sketch, also on the Argand diagram above, the locus of points satisfying the equation $$\arg z = \frac { \pi } { 3 }$$ [1 mark] 12
3. For the complex number \(w\) find the maximum value of \(| w |\) such that $$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$ $$y = \frac { 2 x + 7 } { 3 x + 5 }$$

13

1. Write down the equations of the asymptotes of curve \(C _ { 1 }\) 13 A curve \(C _ { 1 }\) has equation 13
2. On the axes below, sketch the graph of curve \(C _ { 1 }\) Indicate the values of the intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571} 13
3. Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
4. Curve \(C _ { 2 }\) is a reflection of curve \(C _ { 1 }\) in the line \(y = - x\) Find an equation for curve \(C _ { 2 }\) in the form \(y = \mathrm { f } ( x )\)

14 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 2 } + p x + 7 } \quad x \in \mathbb { R }$$ where \(p\) is a constant.
The graph of \(y = \mathrm { f } ( x )\) has only one asymptote.
14

1. Write down the equation of the asymptote.
   14
2. Find the set of possible values of \(p\) □
   14
3. Find the coordinates of the points at which the graph of \(y = \mathrm { f } ( x )\) intersects the axes. \section*{Question 14 continues on the next page} 14
4. \(\quad A\) curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 } { x ^ { 2 } - 3 x + 7 }$$ The curve \(C\) has a local minimum at the point \(M\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-24_371_835_587_605} The line \(y = k\) intersects curve \(C\) 14 (d) (i) Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14 (d) (ii) Hence, find the \(y\)-coordinate of point \(M\)

15 The two values of \(\theta\) that satisfy the equation $$\sinh ^ { 2 } \theta - \sinh \theta - 2 = 0$$ are \(\theta _ { 1 }\) and \(\theta _ { 2 }\) 15

1. Hamzah is asked to find the value of \(\theta _ { 1 } + \theta _ { 2 }\) He writes his answer as follows:
   The quadratic coefficients are \(a = 1 , b = - 1 , c = - 2\) The sum of the roots is \(- \frac { b } { a }\) So \(\theta _ { 1 } + \theta _ { 2 } = - \frac { - 1 } { 1 } = 1\) Explain Hamzah's error.
   [0pt] [1 mark] 15
2. Find the correct value of \(\theta _ { 1 } + \theta _ { 2 }\) Give your answer as a single logarithm. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-28_2492_1721_217_150}