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 2019 June Q14
14
  1. Find \(x\) in terms of \(t\).
    14
  2. State, giving a reason, the type of damping which occurs.
AQA Further Paper 1 2019 June Q15
15 The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \(( r , \theta )\) where \(0 \leq \theta \leq \frac { \pi } { 2 }\)
The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole.
\(T P Q\) is the tangent to the curve at \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-26_624_730_653_653} 15
  1. Show that the gradient of \(T P Q\) is equal to $$\frac { \frac { \mathrm { d } r } { \mathrm {~d} \theta } \sin \theta + r \cos \theta } { \frac { \mathrm {~d} r } { \mathrm {~d} \theta } \cos \theta - r \sin \theta }$$ 15
  2. The curve has polar equation $$r = \mathrm { e } ^ { ( \cot b ) \theta }$$ where \(b\) is a constant such that \(0 < b < \frac { \pi } { 2 }\) Use the result of part (a) to show that the angle between the line \(O P\) and the tangent TPQ does not depend on \(\theta\).
    \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-28_2488_1719_219_150} Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 1 2020 June Q1
1 Which of the integrals below is not an improper integral?
Circle your answer.
\(\int _ { 0 } ^ { \infty } e ^ { - x } d x\)
\(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\)
\(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\)
\(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)
AQA Further Paper 1 2020 June Q2
1 marks
2 Which one of the matrices below represents a rotation of \(90 ^ { \circ }\) about the \(x\)-axis? Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { c c c } 1 & 0 & 0
0 & 1 & 0
0 & 0 & - 1 \end{array} \right]\)
\(\left[ \begin{array} { c c c } - 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\)
\(\left[ \begin{array} { l l l } 1 & 0 & 0
0 & 0 & 1
0 & 1 & 0 \end{array} \right]\)
\(\left[ \begin{array} { c c c } 1 & 0 & 0
0 & 0 & - 1
0 & 1 & 0 \end{array} \right]\)
AQA Further Paper 1 2020 June Q3
3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\). One of the four statements below is incorrect. Which statement is incorrect? Tick ( \(\checkmark\) ) one box.
\(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) □
\(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □
\(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □
\(b = 0 \Rightarrow \alpha = - \beta\) □
AQA Further Paper 1 2020 June Q4
4 （a）Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients．
［4 marks］
4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
堛的 增
4 (b) Find the value of \(p\) and the value of \(r\).
AQA Further Paper 1 2020 June Q5
4 marks
5
  1. Show that the equation of \(H _ { 1 }\) can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where \(q\) and \(r\) are integers.
    5
  2. \(\quad \mathrm { H } _ { 2 }\) is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of \(H _ { 2 }\) onto the graph of \(H _ { 1 }\)
    [0pt] [4 marks]
AQA Further Paper 1 2020 June Q6
2 marks
6 Let \(w\) be the root of the equation \(z ^ { 7 } = 1\) that has the smallest argument \(\alpha\) in the interval \(0 < \alpha < \pi\) 6
  1. Prove that \(w ^ { n }\) is also a root of the equation \(z ^ { 7 } = 1\) for any integer \(n\). 6
  2. Prove that \(1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0\)
    6
  3. Show the positions of \(w , w ^ { 2 } , w ^ { 3 } , w ^ { 4 } , w ^ { 5 }\), and \(w ^ { 6 }\) on the Argand diagram below.
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-08_835_898_1802_571} 6
  4. Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$
AQA Further Paper 1 2020 June Q7
4 marks
7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3
( k - 1 ) x + ( 3 - k ) y + 2 z & = 1
7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7
  1. The planes do not meet at a unique point.
    Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
    7
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes.
    In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.
    [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA Further Paper 1 2020 June Q8
8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where \(k\) is a constant, form an arithmetic sequence. Find the roots of the equation.
AQA Further Paper 1 2020 June Q9
4 marks
9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9
  1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
    Fully justify your answer.
    9
  2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\)
    9
  3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
    \section*{Question 9 continues on the next page} 9
  4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470}
    \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
  5. Fird \(\begin{aligned} & \text { Do not write }
    & \text { outside the } \end{aligned}\)
AQA Further Paper 1 2020 June Q10
10
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
  2. Find the particular solution for which \(y = 0\) when \(x = 3\)
    Give your answer in the form \(y = \mathrm { f } ( x )\)
AQA Further Paper 1 2020 June Q11
5 marks
11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1
5
- 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2
1
- 3 \end{array} \right] = \mathbf { 0 }
& l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3
2
7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2
- 1
3 \end{array} \right] = \mathbf { 0 }
& l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5
12
- 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4
0
9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11
    1. Explain how you know that two of the lines are parallel.
      11
    2. (ii)
      3. Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.
      [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
      11
    3. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection.
      \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}
AQA Further Paper 1 2020 June Q12
6 marks
12
  1. Use the definition of the cosh function to prove that $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) = \ln \left( \frac { x + \sqrt { x ^ { 2 } - a ^ { 2 } } } { a } \right) \quad \text { for } a > 0$$ [6 marks]
    12
  2. The formulae booklet gives the integral of \(\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }\) as $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) \text { or } \ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) + c$$ Ronald says that this contradicts the result given in part (a).
    Explain why Ronald is wrong.
AQA Further Paper 1 2020 June Q13
4 marks
13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart.
\(A B C\) is a horizontal line.
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13
  1. Show that the particle moves with simple harmonic motion.
    13
  2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
    [0pt] [4 marks]
AQA Further Paper 1 2020 June Q14
14
  1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\)
    14
  2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\)
    Do not write
    \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}
AQA Further Paper 1 2021 June Q1
1 Find $$\sum _ { r = 1 } ^ { 20 } \left( r ^ { 2 } - 2 r \right)$$ Circle your answer. 24502660532043680
AQA Further Paper 1 2021 June Q2
2 Given that \(z = 1 - 3 \mathrm { i }\) is one root of the equation \(z ^ { 2 } + p z + r = 0\), where \(p\) and \(r\) are real, find the value of \(r\). Circle your answer.
\(- 8 - 2610\)
AQA Further Paper 1 2021 June Q3
3 The curve \(C\) has polar equation $$r ^ { 2 } \sin 2 \theta = 4$$ Find a Cartesian equation for \(C\).
Circle your answer.
\(y = 2 x\)
\(y = \frac { x } { 2 }\)
\(y = \frac { 2 } { x }\)
\(y = 4 x\)
AQA Further Paper 1 2021 June Q4
4 Show that the solutions to the equation $$3 \tanh ^ { 2 } x - 2 \operatorname { sech } x = 2$$ can be expressed in the form $$x = \pm \ln ( a + \sqrt { b } )$$ where \(a\) and \(b\) are integers to be found.
You may use without proof the result \(\cosh ^ { - 1 } y = \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right)\)
AQA Further Paper 1 2021 June Q5
5 marks
5 The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { c c c } 3 & 2 & - 2
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\) Prove by induction that \(\mathbf { M } ^ { n } = \left[ \begin{array} { c c c } 3 ^ { n } & 3 ^ { n } - 1 & - 3 ^ { n } + 1
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\) for all integers \(n \geq 1\) [5 marks]
AQA Further Paper 1 2021 June Q6
6
  1. Show that the equation $$\left( 2 z - z ^ { * } \right) ^ { * } = z ^ { 2 }$$ has exactly four solutions and state these solutions.
    6
    1. Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry.
      \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-09_842_860_406_589} 6
  3. (ii) Show that the area of this quadrilateral is \(\frac { \sqrt { 15 } } { 2 }\) square units.
AQA Further Paper 1 2021 June Q7
7 The diagram below shows the graph of \(y = \mathrm { f } ( x ) \quad ( - 4 \leq x \leq 4 )\) The graph meets the \(x\)-axis at \(x = 1\) and \(x = 3\)
The graph meets the \(y\)-axis at \(y = 2\)
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-10_846_854_539_593} 7
  1. Sketch the graph of \(y = | \mathrm { f } ( x ) |\) on the axes below.
    Show any axis intercepts.
    \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-10_844_844_1601_598} 7
  2. Sketch the graph of \(y = \frac { 1 } { \mathrm { f } ( x ) }\) on the axes below.
    Show any axis intercepts and asymptotes.
    \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-11_844_846_495_603} 7
  3. Sketch the graph of \(y = \mathrm { f } ( | x | )\) on the axes below.
    Show any axis intercepts.
    \includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-11_848_849_1647_593}
AQA Further Paper 1 2021 June Q8
8 A particle of mass 4 kg moves horizontally in a straight line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The following horizontal forces act on the particle:
  • a constant driving force of magnitude 1.8 newtons
  • another driving force of magnitude \(30 \sqrt { t }\) newtons
  • a resistive force of magnitude \(0.08 v ^ { 2 }\) newtons
When \(t = 70 , v = 54\)
Use Euler's method with a step length of 0.5 to estimate the velocity of the particle after 71 seconds. Give your answer to four significant figures.
AQA Further Paper 1 2021 June Q9
10 marks
9 Use l'Hôpital's rule to show that $$\lim _ { x \rightarrow \infty } \left( x \mathrm { e } ^ { - x } \right) = 0$$ Fully justify your answer.
[0pt] [4 marks]
10
Evaluate the improper integral \(\int _ { 0 } ^ { 8 } \ln x \mathrm {~d
showing the limiting process.
}		[6 marks]
\(11 \quad\) The line \(L _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 2
2
3 \end{array} \right] + \lambda \left[ \begin{array} { c } 2
3
- 1 \end{array} \right]\)
The line \(L _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 6
4
1 \end{array} \right] + \mu \left[ \begin{array} { c } - 2
1
1 \end{array} \right]\)