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AQA Further AS Paper 1 2021 June Q12

5 marks Standard +0.8

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

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

AQA Further AS Paper 1 2021 June Q13

4 marks Moderate -0.5

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

AQA Further AS Paper 1 2021 June Q14

9 marks Standard +0.3

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

Curve $C _ { 2 }$ is a reflection of $C _ { 1 }$ in the line $y = x$ Write down an equation of $C _ { 2 }$ 14
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
Curve $C _ { 4 }$ is a translation of $C _ { 1 }$ The positive $x$-axis and the positive $y$-axis are tangents to $C _ { 4 }$ 14 (c) (i) 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 (c) (ii) Determine the translation vector.
[0pt] [2 marks]
14 (c) (iii) The line $y = m x + c$ is a tangent to both $C _ { 1 }$ and $C _ { 4 }$ Find the value of $m$

AQA Further AS Paper 1 2021 June Q15

8 marks Standard +0.3

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. Show that the two lines intersect.
  [0pt] [3 marks]
  15
  1. (ii) Find the position vector of the point of intersection.
    15
2. 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
3. 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 $0.1 ^ { \circ }$

AQA Further AS Paper 1 2021 June Q16

8 marks Standard +0.3

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

Write down the value of $a$ and the value of $b$ 16
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
Using the values found in part (a), solve the inequality $$\frac { a x } { x + b } \leq 1 - x$$ [4 marks]

AQA Further AS Paper 1 2021 June Q17

12 marks Standard +0.3

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

Write down, in terms of $a$, the polar coordinates of $M$ 17
$\quad N$ is the point on $C _ { 1 }$ that is furthest from the pole $O$ Find, in terms of $a$, the polar coordinates of $N$ 17
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
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}

$\quad$ Find $\operatorname { det } \mathbf { A }$ 6
Find $\mathbf { A } ^ { - 1 }$ 6
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 $\mathbf { M }$

AQA Further AS Paper 1 2022 June Q7

9 marks Standard +0.3

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

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

AQA Further AS Paper 1 2022 June Q8

7 marks Standard +0.3

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

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

AQA Further AS Paper 1 2022 June Q9

5 marks Standard +0.8

Show that, for $r > 0$, $$\ln ( r + 2 ) - \ln r = \ln \left( 1 + \frac { 2 } { r } \right)$$ 9
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.

AQA Further AS Paper 1 2022 June Q10

6 marks Standard +0.3

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

Write down an equation of $E$ 10
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}

AQA Further AS Paper 1 2022 June Q11

4 marks Standard +0.8

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.

AQA Further AS Paper 1 2022 June Q12

6 marks Standard +0.3

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
Sketch, also on the Argand diagram above, the locus of points satisfying the equation $$\arg z = \frac { \pi } { 3 }$$ [1 mark] 12
For the complex number $w$ find the maximum value of $| w |$ such that $$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$ $$y = \frac { 2 x + 7 } { 3 x + 5 }$$

AQA Further AS Paper 1 2022 June Q13

10 marks Standard +0.3

Write down the equations of the asymptotes of curve $C _ { 1 }$ 13 A curve $C _ { 1 }$ has equation 13
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
Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
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 $y = \mathrm { f } ( x )$

AQA Further AS Paper 1 2022 June Q14

15 marks Challenging +1.2

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

Write down the equation of the asymptote.
14
Find the set of possible values of $p$ □
14
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
$\quad A$ 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 $M$

AQA Further AS Paper 1 2022 June Q15

6 marks Standard +0.3

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

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
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}