Questions — AQA Further Paper 1 (97 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 Paper 1 2023 June Q10
10 The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 2 & - 1 & 1
- 1 & - 1 & - 2
1 & 2 & c \end{array} \right]$$ where \(c\) is a real number. 10
  1. The linear transformation T is represented by the matrix \(\mathbf { M }\)
    Show that, for one particular value of \(c\), the image under \(T\) of every point lies in the plane $$x + 5 y + 3 z = 0$$ State the value of \(c\) for which this occurs.
    10
  2. It is given that \(\mathbf { M }\) is a non-singular matrix.
    10
    1. State any restrictions on the value of \(c\)
      10
    2. (iii) Using your answer from part (b)(ii), solve \(\begin{array} { r } 2 x - y + z = - 3
      3. - x - y - 2 z = - 6
      x + 2 y + 4 z = 13 \end{array}\)\(\_\_\_\_\)
AQA Further Paper 1 2023 June Q11
11 The function f is defined by $$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } - 51 x - 45 \quad ( x \in \mathbb { R } )$$ 11
    1. Fully factorise \(\mathrm { f } ( x )\)
      11
  2. (ii) Hence, solve the inequality \(\mathrm { f } ( x ) < 0\)
    11
  3. The graph of \(y = \mathrm { f } ( x )\) is translated by the vector \(\left[ \begin{array} { l } 7
    0 \end{array} \right]\)
    The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = \mathrm { g } ( x )\)
    Solve the inequality \(\mathrm { g } ( x ) \leq 0\)
AQA Further Paper 1 2023 June Q12
12
  1. Starting from the identities for \(\sinh 2 x\) and \(\cosh 2 x\), prove the identity $$\tanh 2 x = \frac { 2 \tanh x } { 1 + \tanh ^ { 2 } x }$$ 12
    1. The function f is defined by $$\mathrm { f } ( x ) = \tanh x \quad ( x > 0 )$$ State the range of f
      12
  3. (ii) Use part (a) and part (b)(i) to prove that \(\tanh 2 x > \tanh x\) if \(x > 0\)
AQA Further Paper 1 2023 June Q13
13 Use l'Hôpital's rule to prove that $$\lim _ { x \rightarrow \pi } \left( \frac { x \sin 2 x } { \cos \left( \frac { x } { 2 } \right) } \right) = - 4 \pi$$
AQA Further Paper 1 2023 June Q14
6 marks
14 The curve \(C\) has polar equation $$r = \frac { 4 } { 5 + 3 \cos \theta } \quad ( - \pi < \theta \leq \pi )$$ 14
  1. Show that \(r\) takes values in the range \(\frac { 1 } { k } \leq r \leq k\), where \(k\) is an integer.
    [0pt] [2 marks] 14
  2. Find the Cartesian equation of \(C\) in the form \(y ^ { 2 } = \mathrm { f } ( x )\) 14
  3. The ellipse \(E\) has equation $$y ^ { 2 } + \frac { 16 x ^ { 2 } } { 25 } = 1$$ Find the transformation that maps the graph of \(E\) onto \(C\)
    [0pt] [4 marks]
    15Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 4 y = \cos 2 x + 5 x\)
AQA Further Paper 1 2023 June Q16
7 marks
16
  1. Show that $$\int _ { 0.5 } ^ { 4 } \frac { 1 } { t } \ln t \mathrm {~d} t = a ( \ln 2 ) ^ { 2 }$$ where \(a\) is a rational number to be found.
    16
  2. A curve \(C\) is defined parametrically for \(t > 0\) by $$x = 2 t \quad y = \frac { 1 } { 2 } t ^ { 2 } - \ln t$$ The arc formed by the graph of \(C\) from \(t = 0.5\) to \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to generate a surface with area \(S\) Find the exact value of \(S\), giving your answer in the form $$S = \pi \left( b + c \ln 2 + d ( \ln 2 ) ^ { 2 } \right)$$ where \(b , c\) and \(d\) are rational numbers to be found.
    [0pt] [7 marks]
    \includegraphics[max width=\textwidth, alt={}]{a9f88195-e545-43f2-a13a-6459d14e1cda-25_2488_1719_219_150}
    Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 1 2024 June Q1
1 The roots of the equation \(20 x ^ { 3 } - 16 x ^ { 2 } - 4 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\)
Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\)
Circle your answer.
\(- \frac { 4 } { 5 }\)
\(- \frac { 1 } { 5 }\)
\(\frac { 1 } { 5 }\)
\(\frac { 4 } { 5 }\)
AQA Further Paper 1 2024 June Q2
1 marks
2 The complex number \(z = e ^ { \frac { i \pi } { 3 } }\)
Which one of the following is a real number?
Circle your answer.
[0pt] [1 mark]
\(z ^ { 4 }\)
\(z ^ { 5 }\)
\(z ^ { 6 }\)
\(z ^ { 7 }\)
AQA Further Paper 1 2024 June Q3
3 The function f is defined by $$f ( x ) = x ^ { 2 } \quad ( x \in \mathbb { R } )$$ Find the mean value of \(\mathrm { f } ( x )\) between \(x = 0\) and \(x = 2\)
Circle your answer.
\(\frac { 2 } { 3 }\)
\(\frac { 4 } { 3 }\)
\(\frac { 8 } { 3 }\)
\(\frac { 16 } { 3 }\)
AQA Further Paper 1 2024 June Q4
4 Which one of the following statements is correct?
Tick ( ✓ ) one box.
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right) = 0\) □
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right) = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-03_110_108_1238_991}
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right) = 2\) □
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right)\) is not defined.
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-03_106_108_1564_991}
AQA Further Paper 1 2024 June Q5
3 marks
5 The points \(A , B\) and \(C\) have coordinates \(A ( 5,3,4 ) , B ( 8 , - 1,9 )\) and \(C ( 12,5,10 )\) The points \(A , B\) and \(C\) lie in the plane \(\Pi\) 5
  1. Find a vector that is normal to the plane \(\Pi\)
    [0pt] [3 marks]
    5
  		2. Find a Cartesian equation of the plane \(\Pi\)
AQA Further Paper 1 2024 June Q6
6 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = u _ { n } + 3 n \end{aligned}$$ Prove by induction that for all integers \(n \geq 1\) $$u _ { n } = \frac { 3 } { 2 } n ^ { 2 } - \frac { 3 } { 2 } n + 1$$
AQA Further Paper 1 2024 June Q8
4 marks
8 The ellipse \(E\) has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 9 } = 1$$ The line with equation \(y = m x + 4\) is a tangent to \(E\)
Without using differentiation show that \(m = \pm \sqrt { 7 }\)
[0pt] [4 marks]
AQA Further Paper 1 2024 June Q9
4 marks
9
  1. It is given that Starting from the exponential definition of the sinh function, show that \(\sinh p = r\) $$p = \ln \left( r + \sqrt { r ^ { 2 } + 1 } \right)$$ Staring fr
    9
  2. Solve the equation $$\cosh ^ { 2 } x = 2 \sinh x + 16$$ Give your answers in logarithmic form.
    [0pt] [4 marks]
    The complex numbers \(z\) and \(w\) are defined by $$\begin{aligned} z & = \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 }
    \text { and } \quad w & = \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \end{aligned}$$ By evaluating the product \(z w\), show that $$\tan \frac { 5 \pi } { 12 } = 2 + \sqrt { 3 }$$
AQA Further Paper 1 2024 June Q11
11
  1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 2 } \tan ^ { - 1 } x \right)\) 11
  2. Hence find \(\int 2 x \tan ^ { - 1 } x \mathrm {~d} x\)
AQA Further Paper 1 2024 June Q12
4 marks
12 The line \(L _ { 1 }\) has equation $$\mathbf { r } = \left[ \begin{array} { l } 4
2
1 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
3
- 1 \end{array} \right]$$ The transformation T is represented by the matrix $$\left[ \begin{array} { c c c } 2 & 1 & 0
3 & 4 & 6
- 5 & 2 & - 3 \end{array} \right]$$ The transformation T transforms the line \(L _ { 1 }\) to the line \(L _ { 2 }\) 12
  1. Show that the angle between \(L _ { 1 }\) and \(L _ { 2 }\) is 0.701 radians, correct to three decimal places.
    [0pt] [4 marks]
    12
  2. Find the shortest distance between \(L _ { 1 }\) and \(L _ { 2 }\)
    Give your answer in an exact form.
AQA Further Paper 1 2024 June Q13
13
  1. Use de Moivre's theorem to show that $$\cos 3 \theta = 4 \cos ^ { 3 } \theta - 3 \cos \theta$$ 13
  2. Use de Moivre's theorem to express \(\sin 3 \theta\) in terms of \(\sin \theta\)
    13
  3. Hence show that $$\cot 3 \theta = \frac { \cot ^ { 3 } \theta - 3 \cot \theta } { 3 \cot ^ { 2 } \theta - 1 }$$
AQA Further Paper 1 2024 June Q14
14 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tanh x = \sinh ^ { 3 } x$$ given that \(y = 3\) when \(x = \ln 2\)
Give your answer in an exact form.
AQA Further Paper 1 2024 June Q15
15 A curve is defined parametrically by the equations $$\begin{array} { l l } x = \frac { 3 } { 2 } t ^ { 3 } + 5 &
y = t ^ { \frac { 9 } { 2 } } & ( t \geq 0 ) \end{array}$$ Show that the arc length of the curve from \(t = 0\) to \(t = 2\) is equal to 26 units.
AQA Further Paper 1 2024 June Q16
16 The curve \(C\) has polar equation \(r = 2 + \tan \theta\) The curve \(C\) meets the line \(\theta = \frac { \pi } { 4 }\) at the point \(A\)
The point \(B\) has polar coordinates \(( 4,0 )\)
The diagram shows part of the curve \(C\), and the points \(A\) and \(B\)
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-22_515_1168_575_427} 16
  1. Show that the area of triangle \(O A B\) is \(3 \sqrt { 2 }\) units.
    16
  2. Find the area of the shaded region.
    Give your answer in an exact form.
AQA Further Paper 1 2024 June Q17
17 By making a suitable substitution, show that $$\int _ { - 2 } ^ { 1 } \sqrt { x ^ { 2 } + 6 x + 8 } d x = 2 \sqrt { 15 } - \frac { 1 } { 2 } \cosh ^ { - 1 } ( 4 )$$
AQA Further Paper 1 2024 June Q18
3 marks
18 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Two light elastic strings each have one end attached to a small ball \(B\) of mass 0.5 kg The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart with \(A\) vertically above \(C\) The whole system is in a thin tube of oil, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-26_439_154_685_927} The string connecting \(A\) and \(B\) has natural length 2 metres, and the tension in this string is \(7 e\) newtons when the extension is \(e\) metres. The string connecting \(B\) and \(C\) has natural length 3 metres, and the tension in this string is \(3 e\) newtons when the extension is \(e\) metres. 18
  1. Find the extension of each string when the system is in equilibrium.
    18
  2. It is known that in a large bath of oil, the oil causes a resistive force of magnitude \(4.5 v\) newtons to act on the ball, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii). 18
    1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards C, and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 9 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 20 x = 0$$ where \(x\) metres is the displacement of the particle below the equilibrium position at time \(t\) seconds after the particle is released.
      [0pt] [3 marks]
      18
  4. (ii) Find \(x\) in terms of \(t\)
    29 18
  5. State one limitation of the model used in part (b)