Questions — AQA Further AS Paper 1 (119 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Further AS Paper 1 2020 June Q11
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}
AQA Further AS Paper 1 2020 June Q12
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 \(4 \leq x \leq 8\)
AQA Further AS Paper 1 2020 June Q13
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}
AQA Further AS Paper 1 2020 June Q14
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 $$\frac { x + 7 } { x + 1 } \leq x + 1$$
AQA Further AS Paper 1 2020 June Q15
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.
AQA Further AS Paper 1 2020 June Q16
4 marks
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.
AQA Further AS Paper 1 2020 June Q17
4 marks
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]
AQA Further AS Paper 1 2020 June Q18
2 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}
AQA Further AS Paper 1 2021 June Q1
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. $$\begin{aligned} & \cos ( - 2 ) + i \sin ( - 2 )
& \cos ( - 1 ) + i \sin ( - 1 )
& \cos ( 1 ) + i \sin ( 1 )
& \cos ( 2 ) + i \sin ( 2 ) \end{aligned}$$ □
□
□
□
AQA Further AS Paper 1 2021 June Q2
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
AQA Further AS Paper 1 2021 June Q3
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. $$\begin{array} { l l } a = \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square
a = \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \square
a = - \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square
a = - \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \end{array}$$
AQA Further AS Paper 1 2021 June Q4
1 marks
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]
\(\left[ \begin{array} { l l } 4 & 6
2 & 5 \end{array} \right]\)
\(\left[ \begin{array} { l l } 6 & 5
4 & 2 \end{array} \right]\)
\(\left[ \begin{array} { l l } 5 & 2
6 & 4 \end{array} \right]\)
\(\left[ \begin{array} { l l } 2 & 4
5 & 6 \end{array} \right]\)
AQA Further AS Paper 1 2021 June Q5
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.
AQA Further AS Paper 1 2021 June Q6
6 Prove the identity $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
AQA Further AS Paper 1 2021 June Q7
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.
AQA Further AS Paper 1 2021 June Q8
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 \(q\)
AQA Further AS Paper 1 2021 June Q9
4 marks
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 $$\sum _ { r = n + 1 } ^ { 5 n } r ( r + 3 )$$ $$\mathbf { A } = \left[ \begin{array} { c c } 3 & i - 1
    i & 2 \end{array} \right]$$
AQA Further AS Paper 1 2021 June Q10
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 \(p\)
AQA Further AS Paper 1 2021 June Q11
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.
AQA Further AS Paper 1 2021 June Q12
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.
AQA Further AS Paper 1 2021 June Q13
4 marks
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
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
    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 \(m\)
AQA Further AS Paper 1 2021 June Q15
3 marks
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
  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 \(0.1 ^ { \circ }\)
AQA Further AS Paper 1 2021 June Q16
4 marks
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]
AQA Further AS Paper 1 2021 June Q17
2 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}