AQA Further Paper 1 (Further Paper 1) 2020 June

1 Which of the integrals below is not an improper integral?
Circle your answer.
\(\int _ { 0 } ^ { \infty } e ^ { - x } d x\)
\(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\)
\(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\)
\(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)

2 Which one of the matrices below represents a rotation of \(90 ^ { \circ }\) about the \(x\)-axis? Circle your answer.
[0pt] [1 mark]
\(\left[ \begin{array} { c c c } 1 & 0 & 0
0 & 1 & 0
0 & 0 & - 1 \end{array} \right]\)
\(\left[ \begin{array} { c c c } - 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\)
\(\left[ \begin{array} { l l l } 1 & 0 & 0
0 & 0 & 1
0 & 1 & 0 \end{array} \right]\)
\(\left[ \begin{array} { c c c } 1 & 0 & 0
0 & 0 & - 1
0 & 1 & 0 \end{array} \right]\)

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 \(\beta > 0\) □
\(b = 0 \Rightarrow \alpha = - \beta\) □

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 \(r\).

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]

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 $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$

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

   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] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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.

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 \(\begin{aligned} & \text { Do not write }
   & \text { outside the } \end{aligned}\)

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 \(y = \mathrm { f } ( x )\)

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. 1. Explain how you know that two of the lines are parallel.
      

      11 (ii) Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures. [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

      11
   2. 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}

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.

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]

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}