AQA Further Paper 1 (Further Paper 1) 2024 June

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.
\(- \frac { 4 } { 5 }\)
\(- \frac { 1 } { 5 }\)
\(\frac { 1 } { 5 }\)
\(\frac { 4 } { 5 }\)

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]
\(z ^ { 4 }\)
\(z ^ { 5 }\)
\(z ^ { 6 }\)
\(z ^ { 7 }\)

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.
\(\frac { 2 } { 3 }\)
\(\frac { 4 } { 3 }\)
\(\frac { 8 } { 3 }\)
\(\frac { 16 } { 3 }\)

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}

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	Find a Cartesian equation of the plane \(\Pi\)

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 \(n \geq 1\) $$u _ { n } = \frac { 3 } { 2 } n ^ { 2 } - \frac { 3 } { 2 } n + 1$$

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]

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 $$\tan \frac { 5 \pi } { 12 } = 2 + \sqrt { 3 }$$

11

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 2 } \tan ^ { - 1 } x \right)\) 11
2. Hence find \(\int 2 x \tan ^ { - 1 } x \mathrm {~d} x\)

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	Find the shortest distance between \(L _ { 1 }\) and \(L _ { 2 }\) Give your answer in an exact form.

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 $$\cot 3 \theta = \frac { \cot ^ { 3 } \theta - 3 \cot \theta } { 3 \cot ^ { 2 } \theta - 1 }$$

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.

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.

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.

17 By making a suitable substitution, show that $$\int _ { - 2 } ^ { 1 } \sqrt { x ^ { 2 } + 6 x + 8 } d x = 2 \sqrt { 15 } - \frac { 1 } { 2 } \cosh ^ { - 1 } ( 4 )$$

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
3. 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)