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 2022 June Q1
1 Which of the following exponential expressions is equivalent to \(2 \sinh x\) ?
Circle your answer.
\(\mathrm { e } ^ { x }\)
\(\mathrm { e } ^ { x } + \mathrm { e } ^ { - x }\)
\(\mathrm { e } ^ { x } - \mathrm { e } ^ { - x }\)
\(\mathrm { e } ^ { - x }\)
AQA Further AS Paper 1 2022 June Q2
1 marks
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]
\(p - p - q - q\)
AQA Further AS Paper 1 2022 June Q3
1 marks
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 \(y = 0\) □
AQA Further AS Paper 1 2022 June Q4
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}
□
AQA Further AS Paper 1 2022 June Q5
3 marks
5 Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) is \(2 + 11 \mathrm { i }\)
[0pt] [3 marks]
AQA Further AS Paper 1 2022 June Q6
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 \(\mathbf { M }\)
AQA Further AS Paper 1 2022 June Q7
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 \(l _ { 2 }\)
AQA Further AS Paper 1 2022 June Q8
3 marks
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 \(\frac { \pi } { 6 }\)
AQA Further AS Paper 1 2022 June Q9
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.
AQA Further AS Paper 1 2022 June Q10
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}
AQA Further AS Paper 1 2022 June Q11
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
1 marks
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 $$| w - 2 \mathrm { i } | \leq 2 \quad \text { and } \quad 0 \leq \arg w \leq \frac { \pi } { 3 }$$ $$y = \frac { 2 x + 7 } { 3 x + 5 }$$
AQA Further AS Paper 1 2022 June Q13
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 \(y = \mathrm { f } ( x )\)
AQA Further AS Paper 1 2022 June Q14
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
    1. Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14
  6. (ii) Hence, find the \(y\)-coordinate of point \(M\)
AQA Further AS Paper 1 2022 June Q15
1 marks
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}
AQA Further AS Paper 1 2024 June Q1
1 Express \(\cosh ^ { 2 } x\) in terms of \(\sinh x\)
Circle your answer.
\(1 + \sinh ^ { 2 } x\)
\(1 - \sinh ^ { 2 } x\)
\(\sinh ^ { 2 } x - 1\)
\(- 1 - \sinh ^ { 2 } x\)
AQA Further AS Paper 1 2024 June Q2
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}
\(2 \pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) d x\) □
\(2 \pi \int _ { 0 } ^ { 5 } ( 2 x + 3 ) ^ { 2 } \mathrm {~d} x\) □
AQA Further AS Paper 1 2024 June Q3
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
AQA Further AS Paper 1 2024 June Q4
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}
\(\frac { x + 9 } { 4 } = \frac { y - 1 } { - 7 } , z = 3\) □
\(\frac { x - 9 } { 4 } = \frac { y + 1 } { - 7 } , z = 3\) □
AQA Further AS Paper 1 2024 June Q5
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.
AQA Further AS Paper 1 2024 June Q6
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}
AQA Further AS Paper 1 2024 June Q7
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.
AQA Further AS Paper 1 2024 June Q8
8
  1. The complex number \(z\) is given by \(z = x + i y\) where \(x , y \in \mathbb { R }\) 8
    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
    1. Find \(w\)
      8
  6. (ii) Calculate the value of \(w ^ { 2 } \left( w ^ { * } \right) ^ { 2 }\)
AQA Further AS Paper 1 2024 June Q9
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}
AQA Further AS Paper 1 2024 June Q10
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
    1. Draw the line \(L\) on Figure 1 10
  4. (ii) Hence, or otherwise, solve the inequality $$\frac { 2 x - 10 } { 3 x - 5 } \leq - \frac { 2 } { 5 } x + 2$$