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AQA Further Paper 1 2023 June Q4
1 marks Moderate -0.5
4 The solution of a second order differential equation is \(\mathrm { f } ( t )\) The differential equation models heavy damping.
Which one of the statements below could be true?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \mathrm { f } ( t ) = 2 \mathrm { e } ^ { - t } \cos ( 3 t ) + 5 \mathrm { e } ^ { - t } \sin ( 3 t ) \\ & \mathrm { f } ( t ) = 3 \mathrm { e } ^ { - t } + 4 t \mathrm { e } ^ { - t } \\ & \mathrm { f } ( t ) = 7 \mathrm { e } ^ { - t } + 2 \mathrm { e } ^ { - 2 t } \\ & \mathrm { f } ( t ) = 8 \mathrm { e } ^ { - t } \cos ( 3 t - 0.1 ) \end{aligned}$$ □
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AQA Further Paper 1 2023 June Q5
6 marks Challenging +1.2
5 The function f is defined by $$f ( r ) = 2 ^ { r } ( r - 2 ) \quad ( r \in \mathbb { Z } )$$ 5
  1. Show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r 2 ^ { r }$$ 5
  2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } r 2 ^ { r } = 2 ^ { n + 1 } ( n - 1 ) + 2$$
AQA Further Paper 1 2023 June Q6
11 marks Standard +0.3
6 The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \frac { 1 } { 10 } \left[ \begin{array} { c c c } a & a & - 6 \\ 0 & 10 & 0 \\ 9 & 14 & - 13 \end{array} \right]$$ where \(a\) is a real number. The vectors \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\), and \(\mathbf { v } _ { 3 }\) are eigenvectors of \(\mathbf { M }\) The corresponding eigenvalues are \(\lambda _ { 1 } , \lambda _ { 2 }\), and \(\lambda _ { 3 }\) respectively.
It is given that \(\lambda _ { 2 } = 1\) and \(\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 3 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]\) and \(\mathbf { v } _ { 3 } = \left[ \begin{array} { l } c \\ 0 \\ 1 \end{array} \right]\),
where \(c\) is an integer. 6
    1. Find the value of \(\lambda _ { 1 }\) 6
  2. (ii) Find the value of \(a\) 6
  3. Find the integer \(c\) and the value of \(\lambda _ { 3 }\) 6
  4. Find matrices \(\mathbf { U } , \mathbf { D }\) and \(\mathbf { U } ^ { - 1 }\), such that \(\mathbf { D }\) is diagonal and \(\mathbf { M } = \mathbf { U D U } ^ { - 1 }\)
AQA Further Paper 1 2023 June Q7
5 marks Standard +0.8
7 The function f is defined by $$f ( x ) = \left| \sin x + \frac { 1 } { 2 } \right| \quad ( 0 \leq x \leq 2 \pi )$$ Find the set of values of \(x\) for which $$f ( x ) \geq \frac { 1 } { 2 }$$ Give your answer in set notation.
AQA Further Paper 1 2023 June Q8
5 marks Standard +0.3
8 The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { \sin x } \quad ( 0 \leq x \leq 2 \pi )$$ The diagram below shows the graph of \(y = \mathrm { g } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-09_369_593_548_721} 8
  1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = \mathrm { g } ( x )\), giving your answers in exact form. 8
  2. Use Simpson's rule with 3 ordinates to estimate $$\int _ { 0 } ^ { \pi } g ( x ) d x$$ giving your answer to two decimal places.
    8
  3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b).
AQA Further Paper 1 2023 June Q9
9 marks Standard +0.3
9 The position vectors of the points \(A , B\) and \(C\) are $$\begin{aligned} & \mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } \\ & \mathbf { b } = - \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k } \\ & \mathbf { c } = - 2 \mathbf { j } \end{aligned}$$ respectively.
9
  1. Find the area of the triangle \(A B C\) 9
  2. The points \(A , B\) and \(C\) all lie in the plane \(\Pi\) Find an equation of the plane \(\Pi\), in the form \(\mathbf { r } \cdot \mathbf { n } = d\) \(\mathbf { 9 ( c ) } \quad\) The point \(P\) has position vector \(\mathbf { p } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) Find the exact distance of \(P\) from \(\Pi\)
AQA Further Paper 1 2023 June Q10
12 marks Standard +0.8
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
7 marks Standard +0.3
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
6 marks Standard +0.3
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
5 marks Standard +0.3
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
10 marks Challenging +1.3
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
11 marks Challenging +1.2
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 marks Easy -1.2
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 Easy -1.2
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
1 marks Easy -1.2
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
1 marks Standard +0.3
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
5 marks Standard +0.3
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
4 marks Standard +0.3
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 Standard +0.8
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
8 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
5 marks Standard +0.3
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
10 marks Challenging +1.2
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
9 marks Standard +0.3
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
7 marks Challenging +1.2
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
5 marks Standard +0.8
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.