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AQA Further Paper 1 2024 June Q16
9 marks
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
7 marks Challenging +1.8
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
12 marks Challenging +1.2
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)
AQA Further Paper 2 2019 June Q1
1 marks Easy -1.2
1 Given that \(z\) is a complex number, and that \(z ^ { * }\) is the complex conjugate of \(z\), which of the following statements is not always true? Circle your answer.
[0pt] [1 mark] $$\left( z ^ { * } \right) ^ { * } = z \quad z z ^ { * } = | z | ^ { 2 } \quad ( - z ) ^ { * } = - \left( z ^ { * } \right) \quad z - z ^ { * } = z ^ { * } - z$$
AQA Further Paper 2 2019 June Q2
1 marks Moderate -0.8
2 Which of the straight lines given below is an asymptote to the curve $$y = \frac { a x ^ { 2 } } { x - 1 }$$ where \(a\) is a non-zero constant? Circle your answer.
[0pt] [1 mark] \(y = a x + a\) \(y = a x\) \(y = a x - a\) \(y = a\)
AQA Further Paper 2 2019 June Q3
1 marks Moderate -0.8
3 The set \(\mathcal { A }\) is defined by \(\mathcal { A } = \{ x : - \sqrt { } 2 < x < 0 \} \cup \{ x : 0 < x < \sqrt { } 2 \}\) Which of the inequalities given below has \(\mathcal { A }\) as its solution?
Circle your answer.
[0pt] [1 mark] \(\left| x ^ { 2 } - 1 \right| > 1\) \(\left| x ^ { 2 } - 1 \right| \geq 1\) \(\left| x ^ { 2 } - 1 \right| < 1\) \(\left| x ^ { 2 } - 1 \right| \leq 1\)
AQA Further Paper 2 2019 June Q4
3 marks Standard +0.3
4 The positive integer \(k\) is such that $$\sum _ { r = 1 } ^ { k } ( 3 r - k ) = 90$$ Find the value of \(k\).
AQA Further Paper 2 2019 June Q5
4 marks Challenging +1.2
5 A curve has equation \(y = \cosh x\) Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to \(\sinh b - \sinh a\) \(6 \quad\) A circle \(C\) in the complex plane has equation \(| z - 2 - 5 \mathrm { i } | = a\) The point \(z _ { 1 }\) on \(C\) has the least argument of any point on \(C\), and \(\arg \left( z _ { 1 } \right) = \frac { \pi } { 4 }\) Prove that \(a = \frac { 3 \sqrt { } 2 } { 2 }\)
AQA Further Paper 2 2019 June Q7
6 marks Standard +0.3
7 The points \(A , B\) and \(C\) have coordinates \(A ( 4,5,2 ) , B ( - 3,2 , - 4 )\) and \(C ( 2,6,1 )\) 7
  1. Use a vector product to show that the area of triangle \(A B C\) is \(\frac { 5 \sqrt { 11 } } { 2 }\) [0pt] [4 marks]
    7
  2. The points \(A , B\) and \(C\) lie in a plane.
    Find a vector equation of the plane in the form r.n \(= k\) 7
  3. Hence find the exact distance of the plane from the origin.
AQA Further Paper 2 2019 June Q8
9 marks Challenging +1.2
8
  1. The line \(y = m x\) is a tangent to \(P _ { 2 }\) Prove that \(m = \pm \sqrt { \frac { a } { b } }\) Solutions using differentiation will be given no marks.
    8
  2. The line \(y = \sqrt { \frac { a } { b } } x\) meets \(P _ { 2 }\) at the point \(D\).
    The finite region \(R\) is bounded by the \(x\)-axis, \(P _ { 2 }\) and a line through \(D\) perpendicular to the \(x\)-axis. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
    Find, in terms of \(a\) and \(b\), the volume of this solid.
    Fully justify your answer.
  3. Find the eigenvalues and corresponding eigenvectors of the matrix
AQA Further Paper 2 2019 June Q9
13 marks Standard +0.3
9
  1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { 5 } & \frac { 2 } { 5 } \\ \frac { - 3 } { 5 } & \frac { 13 } { 10 } \end{array} \right]$$ 9
  2. Find matrices \(\mathbf { U }\) and \(\mathbf { D }\) such that \(\mathbf { D }\) is a diagonal matrix and \(\mathbf { M } = \mathbf { U D U } ^ { - 1 }\) 9
  3. Given that \(\mathbf { M } ^ { n } \rightarrow \mathbf { L }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { L }\).
    [0pt] [4 marks]
    9
  4. The transformation represented by \(\mathbf { L }\) maps all points onto a line. Find the equation of this line.
    \begin{center} \begin{tabular}{ | l | }
AQA Further Paper 2 2019 June Q10
7 marks Standard +0.3
10
- \(\begin{array} { c } \text { Prove by induction that } \mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n \text { is divisible by } 6 \text { for all integers } n \geq 1 \\ \text { [7 marks] } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \end{array}\) -
\end{tabular} \end{center}
\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-13_2488_1716_219_153}
AQA Further Paper 2 2019 June Q11
8 marks Challenging +1.2
11 The line \(L _ { 1 }\) has equation $$\frac { x - 2 } { 3 } = \frac { y + 4 } { 8 } = \frac { 4 z - 5 } { 5 }$$ The line \(L _ { 2 }\) has equation $$\left( \mathbf { r } - \left[ \begin{array} { c } - 2 \\ 0 \\ 3 \end{array} \right] \right) \times \left[ \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right] = \mathbf { 0 }$$ Find the shortest distance between the two lines, giving your answer to three significant figures.
AQA Further Paper 2 2019 June Q12
5 marks Challenging +1.2
12 Abel and Bonnie are trying to solve this mathematical problem: $$\begin{gathered} z = 2 - 3 \mathrm { i } \text { is a root of the equation } \\ 2 z ^ { 3 } + m z ^ { 2 } + p z + 91 = 0 \end{gathered}$$ Find the value of \(m\) and the value of \(p\). Abel says he has solved the problem.
Bonnie says there is not enough information to solve the problem.
12
  1. Abel's solution begins as follows: Since \(z = 2 - 3 \mathrm { i }\) is a root of the equation, \(z = 2 + 3 \mathrm { i }\) is another root. State one extra piece of information about \(m\) and \(p\) which could be added to the problem to make the beginning of Abel's solution correct.
    12
  		2. Prove that Bonnie is right.
    13(a) Explain why \(\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) is an improper integral.
    [1 mark]
    13(b) Evaluate \(\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x\)
    Show the limiting process.
    [9 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    		\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-18_97_150_215_1884}
AQA Further Paper 2 2019 June Q14
12 marks Challenging +1.8
14
  1. Use the method of differences to show that $$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$ where \(a , b\) and \(c\) are integers.
    Question 14 continues on the next page 14
  2. Show that, for any number \(k\) greater than \(\frac { 12 } { 5 }\), if the difference between \(\frac { 5 } { 12 }\) and \(S _ { n }\) is less than \(\frac { 1 } { k }\), then $$n > \frac { k - 5 + \sqrt { k ^ { 2 } + 1 } } { 2 }$$
AQA Further Paper 2 2019 June Q15
14 marks Challenging +1.8
15
  1. Find the value of \(r\). 15
  2. Show that \(\mu = 3\) 15
  3. Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\).
    [0pt] [9 marks]
    \includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-27_2493_1721_214_150}
    Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 2 2020 June Q1
1 marks Standard +0.3
1 Three of the four expressions below are equivalent to each other.
Which of the four expressions is not equivalent to any of the others? Circle your answer. \(\mathbf { a } \times ( \mathbf { a } + \mathbf { b } )\) \(( \mathbf { a } + \mathbf { b } ) \times \mathbf { b }\) \(( \mathbf { a } - \mathbf { b } ) \times \mathbf { b }\) \(\mathbf { a } \times ( \mathbf { a } - \mathbf { b } )\)
AQA Further Paper 2 2020 June Q2
1 marks Moderate -0.5
2 Given that arg \(( a + b \mathrm { i } ) = \varphi\), where \(a\) and \(b\) are positive real numbers and \(0 < \varphi < \frac { \pi } { 2 }\), three of the following four statements are correct. Which statement is not correct? Tick \(( \checkmark )\) one box. $$\begin{aligned} & \arg ( - a - b \mathrm { i } ) = \pi - \varphi \\ & \arg ( a - b \mathrm { i } ) = - \varphi \\ & \arg ( b + a \mathrm { i } ) = \frac { \pi } { 2 } - \varphi \\ & \arg ( b - a \mathrm { i } ) = \varphi - \frac { \pi } { 2 } \end{aligned}$$
AQA Further Paper 2 2020 June Q3
1 marks Moderate -0.8
3 Find the gradient of the tangent to the curve $$y = \sin ^ { - 1 } x$$ at the point where \(x = \frac { 1 } { 5 }\) Circle your answer.
\(\frac { 5 \sqrt { 6 } } { 12 }\)\(\frac { 2 \sqrt { 6 } } { 5 }\)\(\frac { 4 \sqrt { 3 } } { 25 }\)\(\frac { 25 } { 24 }\)
AQA Further Paper 2 2020 June Q4
3 marks Standard +0.3
4 The matrices A and B are defined as follows: $$\begin{aligned} & \mathbf { A } = \left[ \begin{array} { l l } x + 1 & 2 \\ x + 2 & - 3 \end{array} \right] \\ & \mathbf { B } = \left[ \begin{array} { c c } x - 4 & x - 2 \\ 0 & - 2 \end{array} \right] \end{aligned}$$ Show that there is a value of \(x\) for which \(\mathbf { A B } = k \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer to be found.
AQA Further Paper 2 2020 June Q5
5 marks Moderate -0.5
5 Solve the inequality $$\frac { 2 x + 3 } { x - 1 } \leq x + 5$$
AQA Further Paper 2 2020 June Q6
5 marks Challenging +1.2
6 Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers.
AQA Further Paper 2 2020 June Q7
5 marks Challenging +1.2
7 The diagram shows part of the graph of \(y = \cos ^ { - 1 } x\) The diagram shows part of the graph of \(y = \cos ^ { - 1 } x\) \includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-07_689_958_358_539} The finite region enclosed by the graph of \(y = \cos ^ { - 1 } x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places.
AQA Further Paper 2 2020 June Q8
9 marks Challenging +1.8
8
  1. \(\quad\) Factorise \(\left| \begin{array} { c c c } 2 a + b + x & x + b & x ^ { 2 } + b ^ { 2 } \\ 0 & a & - a ^ { 2 } \\ a + b & b & b ^ { 2 } \end{array} \right|\) as fully as possible.
    8
  2. The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c c } 13 + x & x + 3 & x ^ { 2 } + 9 \\ 0 & 5 & - 25 \\ 8 & 3 & 9 \end{array} \right]$$ Under the transformation represented by \(\mathbf { M }\), a solid of volume \(0.625 \mathrm {~m} ^ { 3 }\) becomes a solid of volume \(300 \mathrm {~m} ^ { 3 }\) Use your answer to part (a) to find the possible values of \(x\).
    Use \(\mathbf { C }\) to show that \(\cos \frac { \pi } { 12 }\) can be written in the form \(\frac { \sqrt { \sqrt { m } + n } } { 2 }\), where \(m\) and \(n\) are integers.
AQA Further Paper 2 2020 June Q10
6 marks Standard +0.8
10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$ Prove by induction that, for all integers \(n \geq 1\), $$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$