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 2023 June Q12
12
  1. Show that \(( 1 + i ) ^ { 4 } = - 4\)
    12
  2. The function f is defined by $$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$ 12
    1. Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
      12
  4. (ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
    12
  5. (iii) One of the linear factors of \(\mathrm { f } ( \mathrm { z } )\) is $$( z - ( 1 + i ) )$$ Write down another linear factor and hence, or otherwise, find a quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12
  6. (iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12
  7. (v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.
AQA Further AS Paper 1 2023 June Q13
4 marks
13
  1. Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ [4 marks]
    13
  2. Hence, or otherwise, write down a factorised expression for the sum of the first \(2 n\) squares $$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
  3. Use the formula in part (a) to write down a factorised expression for the sum of the first \(n\) even squares $$2 ^ { 2 } + 4 ^ { 2 } + 6 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }$$ 13
  4. Hence, or otherwise, show that the sum of the first \(n\) odd squares is $$a n ( b n - 1 ) ( b n + 1 )$$ where \(a\) and \(b\) are rational numbers to be determined.
AQA Further AS Paper 1 2023 June Q14
14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers \(a\) and \(b\)
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AQA Further AS Paper 1 2019 June Q1
1 Which of the following matrices is an identity matrix?
Circle your answer. $$\left[ \begin{array} { l l } 1 & 1
1 & 1 \end{array} \right] \quad \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 0
0 & 0 \end{array} \right]$$
AQA Further AS Paper 1 2019 June Q2
1 marks
2 Which of the following expressions is the determinant of the matrix \(\left[ \begin{array} { l l } a & 2
b & 5 \end{array} \right]\) ?
Circle your answer.
\(5 a - 2 b\)
\(2 a - 5 b\)
\(5 b - 2 a\)
\(2 b - 5 a\)
\(3 \quad\) Point \(P\) has polar coordinates \(\left( 2 , \frac { 2 \pi } { 3 } \right)\).
Which of the following are the Cartesian coordinates of \(P\) ?
Circle your answer.
[0pt] [1 mark]
\(( 1 , - \sqrt { 3 } )\)
\(( - \sqrt { 3 } , 1 )\)
\(( \sqrt { 3 } , - 1 )\)
\(( - 1 , \sqrt { 3 } )\) $$r = \frac { k } { \sin \theta }$$ where \(k\) is a positive constant.
AQA Further AS Paper 1 2019 June Q4
4
  1. Sketch \(L\). The line \(L\) has polar equation 4 The line \(L\) has polar equation $$r = \frac { k } { \sin \theta }$$ where \(k\) is a positive constant.
    Sketch \(L\).
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-03_94_716_1037_662} 4
  2. State the minimum distance between \(L\) and the point \(O\).
AQA Further AS Paper 1 2019 June Q5
2 marks
5
  1. Write down the equations of the asymptotes of \(H\). 5
  2. Sketch the hyperbola \(H\) on the axes below, indicating the coordinates of any points of intersection with the coordinate axes. The asymptotes have already been drawn.
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-04_693_798_1306_623} 5
  3. The finite region bounded by \(H\), the positive \(x\)-axis, the positive \(y\)-axis and the line \(y = a\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Show that the volume of the solid generated is \(m a ^ { 3 }\), where \(m = 3.40\) correct to three significant figures.
AQA Further AS Paper 1 2019 June Q6
4 marks
6
  1. On the axes provided, sketch the graph of $$x = \cosh ( y + b )$$ where \(b\) is a positive constant.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-06_1148_1317_1347_358} 6
  2. Determine the minimum distance between the graph of \(x = \cosh ( y + b )\) and the \(y\)-axis.
AQA Further AS Paper 1 2019 June Q7
7
  1. Show that $$\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { A } { r ^ { 2 } - 1 }$$ where \(A\) is a constant to be found. 7
  2. Hence use the method of differences to show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } \equiv \frac { a n ^ { 2 } + b n + c } { 4 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be found.
AQA Further AS Paper 1 2019 June Q8
2 marks
8 Given that \(z _ { 1 } = 2 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)\) and \(z _ { 2 } = 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)\)
8
  1. Find the value of \(\left| z _ { 1 } z _ { 2 } \right|\) 8
  2. Find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) 8
  3. Sketch \(z _ { 1 }\) and \(z _ { 2 }\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively.
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-10_764_869_1546_587} 8
  4. A third complex number \(w\) satisfies both \(| w | = 2\) and \(- \pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(P \widehat { R } Q\). Fully justify your answer.
AQA Further AS Paper 1 2019 June Q9
9
  1. Saul is solving the equation $$2 \cosh x + \sinh ^ { 2 } x = 1$$ He writes his steps as follows: $$\begin{aligned} 2 \cosh x + \sinh ^ { 2 } x & = 1
    2 \cosh x + 1 - \cosh ^ { 2 } x & = 1
    2 \cosh x - \cosh ^ { 2 } x & = 0
    \cosh x \neq 0 \therefore 2 - \cosh x & = 0
    \cosh x & = 2
    x & = \pm \cosh ^ { - 1 } ( 2 ) \end{aligned}$$ Identify and explain the error in Saul's method. 9
  2. Anna is solving the different equation
    g (b) Anna is solving the different equation $$\sinh ^ { 2 } ( 2 x ) - 2 \cosh ( 2 x ) = 1$$ and finds the correct answers in the form \(x = \frac { 1 } { p } \cosh ^ { - 1 } ( q + \sqrt { r } )\), where \(p , q\) and \(r\) are integers. Find the possible values of \(p , q\) and \(r\).
    Fully justify your answer.
AQA Further AS Paper 1 2019 June Q10
3 marks
10
  1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(\mathrm { e } ^ { x }\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). 10
  2. Hence find a trigonometric function for which the first three terms of its Maclaurin series are the same as the first three terms of the Maclaurin series for cosh (ix).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-15_2488_1716_219_153}
AQA Further AS Paper 1 2019 June Q11
11
  1. Curve \(C\) has equation $$y = \frac { x ^ { 2 } + p x - q } { x ^ { 2 } - r }$$ where \(p , q\) and \(r\) are positive constants.
    Write down the equations of its asymptotes.
AQA Further AS Paper 1 2019 June Q12
12 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 2
0 & 3 \end{array} \right]$$ 12
  1. Prove by induction that, for all integers \(n \geq 1\), $$\mathbf { A } ^ { n } = \left[ \begin{array} { c c } 1 & 3 ^ { n } - 1
    0 & 3 ^ { n } \end{array} \right]$$ 12
  2. Find all invariant lines under the transformation matrix \(A\). Fully justify your answer.
    12
  3. Find a line of invariant points under the transformation matrix \(\mathbf { A }\).
AQA Further AS Paper 1 2019 June Q13
13 Line \(l _ { 1 }\) has Cartesian equation $$x - 3 = \frac { 2 y + 2 } { 3 } = 2 - z$$ 13
  1. Write the equation of line \(l _ { 1 }\) in the form $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$$ where \(\lambda\) is a parameter and \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.
    13
  2. Line \(l _ { 2 }\) passes through the points \(P ( 3,2,0 )\) and \(Q ( n , 5 , n )\), where \(n\) is a constant.
    13
    1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are not perpendicular.
      13
    2. (ii) Explain briefly why lines \(l _ { 1 }\) and \(l _ { 2 }\) cannot be parallel.
      3. 13
    3. (iii) Given that \(\theta\) is the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\), show that
      4. \(\cos \theta = \frac { p } { \sqrt { 34 n ^ { 2 } + q n + 306 } }\)
      where \(p\) and \(q\) are constants to be found.
AQA Further AS Paper 1 2019 June Q14
14 The graph of \(y = x ^ { 3 } - 3 x\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-22_718_771_370_632} The two stationary points have \(x\)-coordinates of - 1 and 1
The cubic equation $$x ^ { 3 } - 3 x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha , \beta\) and \(\gamma\).
The roots \(\alpha\) and \(\beta\) are not real.
14
  1. Explain why \(\alpha + \beta = - \gamma\)
    14
  2. Find the set of possible values for the real constant \(p\).
    14
  3. \(\quad \mathrm { f } ( x ) = 0\) is a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\)
    14
    1. Show that the constant term of \(\mathrm { f } ( x )\) is \(p + 2\)
      14
  5. (ii) Write down the \(x\)-coordinates of the stationary points of \(y = \mathrm { f } ( x )\)
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-24_2488_1719_219_150} Question number Additional page, if required.
    Write the question numbers in the left-hand margin. Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further AS Paper 1 2020 June Q1
1 Express the complex number \(1 - \mathrm { i } \sqrt { 3 }\) in modulus-argument form.
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) & \square
2 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) & \square
2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) & \square
2 \left( \cos \left( - \frac { 2 \pi } { 3 } \right) + i \sin \left( - \frac { 2 \pi } { 3 } \right) \right) \end{array}$$
AQA Further AS Paper 1 2020 June Q2
2 Given that \(1 - \mathrm { i }\) is a root of the equation \(z ^ { 3 } - 3 z ^ { 2 } + 4 z - 2 = 0\), find the other two roots. Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & - 1 + i \text { and } - 1
& 1 + i \text { and } 1
& - 1 + i \text { and } 1
& 1 + i \text { and } - 1 \end{aligned}$$ □
□
□
□
AQA Further AS Paper 1 2020 June Q3
3 Given \(( x - 1 ) ( x - 2 ) ( x - a ) < 0\) and \(a > 2\) Find the set of possible values of \(x\).
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \{ x : x < 1 \} \cup \{ x : 2 < x < a \}
& \{ x : 1 < x < 2 \} \cup \{ x : x > a \}
& \{ x : x < - a \} \cup \{ x : - 2 < x < - 1 \}
& \{ x : - a < x < - 2 \} \cup \{ x : x > - 1 \} \end{aligned}$$
AQA Further AS Paper 1 2020 June Q4
2 marks
4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that $$\mathbf { A } = \left[ \begin{array} { c c c } 2 & a & 3
0 & - 2 & 1 \end{array} \right] \quad \text { and } \quad \mathbf { B } = \left[ \begin{array} { c c } 1 & - 3
- 2 & 4 a
0 & 5 \end{array} \right]$$ 4
  1. Find the product \(\mathbf { A B }\) in terms of \(a\).
    [0pt] [2 marks]
    4
  2. Find the determinant of \(\mathbf { A B }\) in terms of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-04_31_31_513_367}
    "
    □ \(\quad \mathbf { A } = \left[ \begin{array} { c c c } 2 & a & 3
    0 & - 2 & 1 \end{array} \right]\) and \(\quad \mathbf { B } = \left[ \begin{array} { c c } 1 & - 3
    - 2 & 4 a
    0 & 5 \end{array} \right]\)
    \(\mathbf { 4 }\) (a) Find the product \(\mathbf { A B }\) in terms of \(a\). 4
  3. Show that \(\mathbf { A B }\) is singular when \(a = - 1\)
AQA Further AS Paper 1 2020 June Q5
1 marks
5
  1. Show that $$r ^ { 2 } ( r + 1 ) ^ { 2 } - ( r - 1 ) ^ { 2 } r ^ { 2 } = p r ^ { 3 }$$ where \(p\) is an integer to be found.
    [0pt] [1 mark]
    5
  2. Hence use the method of differences to show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
AQA Further AS Paper 1 2020 June Q6
2 marks
6 Anna has been asked to describe the transformation given by the matrix $$\left[ \begin{array} { c c c } 1 & 0 & 0
0 & - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
0 & \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$ She writes her answer as follows: The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where $$\begin{gathered} \sin \theta = \frac { 1 } { 2 } \quad \text { and } \quad - \sin \theta = - \frac { 1 } { 2 }
\theta = 30 ^ { \circ } \end{gathered}$$ Identify and correct the error in Anna's work.
[0pt] [2 marks]
\(7 \quad\) Prove by induction that, for all integers \(n \geq 1\), the expression \(7 ^ { n } - 3 ^ { n }\) is divisible by 4
AQA Further AS Paper 1 2020 June Q8
8
  1. Prove that
    \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\)
    8
  2. Prove that the graphs of $$y = \sinh x \quad \text { and } \quad y = \cosh x$$ do not intersect.
AQA Further AS Paper 1 2020 June Q9
9 The quadratic equation \(2 x ^ { 2 } + p x + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\). 9
    1. Write down the value of \(\alpha \beta\). 9
  2. (ii) Express \(\alpha + \beta\) in terms of \(p\). 9
  3. Hence find \(( \alpha - \beta ) ^ { 2 }\) in terms of \(p\).
    9
  4. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\)
AQA Further AS Paper 1 2020 June Q10
10
  1. Show that the equation $$y = \frac { 3 x - 5 } { 2 x + 4 }$$ can be written in the form $$( x + a ) ( y + b ) = c$$ where \(a\), \(b\) and \(c\) are integers to be found.
    10
  2. Write down the equations of the asymptotes of the graph of $$y = \frac { 3 x - 5 } { 2 x + 4 }$$ 10
  3. Sketch, on the axes provided, the graph of $$y = \frac { 3 x - 5 } { 2 x + 4 }$$
    \includegraphics[max width=\textwidth, alt={}]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-15_1104_1115_1439_466}