AQA Further Paper 1 (Further Paper 1) Specimen

1
- 1
1 \end{array} \right] \quad \left[ \begin{array} { l } 3
0

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0
2 \end{array} \right] \quad \left[ \begin{array} { c } 5
- 1
3 \end{array} \right] \quad \left[ \begin{array} { l } 2
1
1 \end{array} \right]$$ 2 Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\)
[0pt] [2 marks]
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1. Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers \(A\) and \(B\)
   3
2. Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$

5
- 1
3 \end{array} \right] \quad \left[ \begin{array} { l } 2
1
1 \end{array} \right]$$ 2 Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\)
[0pt] [2 marks]
3

1. Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers \(A\) and \(B\)
   3
2. Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$ 4 A student states that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos x + \sin x } { \cos x - \sin x } \mathrm {~d} x\) is not an improper integral because \(\frac { \cos x + \sin x } { \cos x - \sin x }\) is defined at both \(x = 0\) and \(x = \frac { \pi } { 2 }\) Assess the validity of the student's argument.
   [0pt] [2 marks]
   \(5 \quad \mathrm { p } ( z ) = z ^ { 4 } + 3 z ^ { 2 } + a z + b , a \in \mathbb { R } , b \in \mathbb { R }\)
   \(2 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { p } ( \mathrm { z } ) = 0\) 5
3. Express \(\mathrm { p } ( z )\) as a product of quadratic factors with real coefficients.
   5
4. Solve the equation \(\mathrm { p } ( z ) = 0\).

6

1. Obtain the general solution of the differential equation $$\tan x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \sin x \tan x$$ where \(0 < x < \frac { \pi } { 2 }\)
   [0pt] [5 marks]
   6
2. Hence find the particular solution of this differential equation, given that \(y = \frac { 1 } { 2 \sqrt { 2 } }\)
   when \(x = \frac { \pi } { 4 }\)
   [0pt] [2 marks]

7 Three planes have equations, $$\begin{gathered} x - y + k z = 3
k x - 3 y + 5 z = - 1
x - 2 y + 3 z = - 4 \end{gathered}$$ Where \(k\) is a real constant. The planes do not meet at a unique point. 7

1. Find the possible values of \(k\) 7
2. There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of \(k\) Fully justify your answer.
   [0pt] [5 marks]
   7
3. Given further that the equations of the planes form a consistent system, find the solution of the system of equations.

8 A curve has equation $$y = \frac { 5 - 4 x } { 1 + x }$$ 8

1. Sketch the curve.
   \includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-10_1205_1219_886_360} 8
2. Hence sketch the graph of \(y = \left| \frac { 5 - 4 x } { 1 + x } \right|\).
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-11_1203_1202_641_331}

9 A line has Cartesian equations \(x - p = \frac { y + 2 } { q } = 3 - z\) and a plane has
equation r. \(\left[ \begin{array} { r } 1
- 1
- 2 \end{array} \right] = - 3\) 9

1. In the case where the plane fully contains the line, find the values of \(p\) and \(q\).
   [0pt] [3 marks]
   9
2. In the case where the line intersects the plane at a single point, find the range of values of \(p\) and \(q\).
   [0pt] [3 marks]
   9
3. In the case where the angle \(\theta\) between the line and the plane satisfies \(\sin \theta = \frac { 1 } { \sqrt { 6 } }\) and the line intersects the plane at \(z = 0\) 9
4. 1. Find the value of \(q\).
      [0pt] [4 marks]
      9
5. (ii) Find the value of \(p\).

10 The curve, \(C\), has equation \(y = \frac { x } { \cosh x }\)
10

1. Show that the \(x\)-coordinates of any stationary points of \(C\) satisfy the equation \(\tanh x = \frac { 1 } { x }\)
   [0pt] [3 marks] 10
2. 1. Sketch the graphs of \(y = \tanh x\) and \(y = \frac { 1 } { x }\) on the axes below.
      [0pt] [2 marks]
      \includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-14_1151_1226_1461_358} 10
3. (ii) Hence determine the number of stationary points of the curve \(C\). 10
4. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 0\) at each of the stationary points of the curve \(C\).
   [0pt] [4 marks]

11

1. Prove that \(\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } \equiv 2 \operatorname { coth } \theta\) Explicitly state any hyperbolic identities that you use within your proof.
   [0pt] [4 marks] LL
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   L
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2. Solve \(\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } = 4\) giving your answer in an exact form.
   [0pt] [2 marks]

12 The function \(\mathrm { f } ( x ) = \cosh ( \mathrm { i } x )\) is defined over the domain \(\{ x \in \mathbb { R } : - a \pi \leq x \leq a \pi \}\), where \(a\) is a positive integer. By considering the graph of \(y = [ f ( x ) ] ^ { n }\), find the mean value of \([ f ( x ) ] ^ { n }\), when \(n\) is an odd positive integer. Fully justify your answer.
[0pt] [3 marks]

13 Given that \(\mathbf { M } = \left[ \begin{array} { l l l } 1 & 1 & 1
1 & 1 & 1
1 & 1 & 1 \end{array} \right]\), prove that \(\mathbf { M } ^ { n } = \left[ \begin{array} { l l l } 3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 }
3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 }
3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 } \end{array} \right]\) for all \(n \in \mathbb { N }\)
[0pt] [5 marks] LL LL L
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14 A particle, \(P\), of mass \(M\) is released from rest and moves along a horizontal straight line through a point \(O\). When \(P\) is at a displacement of \(x\) metres from \(O\), moving with a speed \(v \mathrm {~ms} ^ { - 1 }\), a force of magnitude \(| 8 M x |\) acts on the particle directed towards \(O\). A resistive force, of magnitude \(4 M v\), also acts on \(P\). 14

1. Initially \(P\) is held at rest at a displacement of 1 metre from \(O\). Describe completely the motion of \(P\) after it is released. Fully justify your answer.
   [0pt] [8 marks]
   14
2. It is decided to alter the resistive force so that the motion of \(P\) is critically damped. Determine the magnitude of the resistive force that will produce critically damped motion.
   [0pt] [4 marks]

15 An isolated island is populated by rabbits and foxes. At time \(t\) the number of rabbits is \(x\) and the number of foxes is \(y\). It is assumed that:

• The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time.
• If there were no foxes present, the number of rabbits would increase by \(120 \%\) in a unit period of time.
• When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to \(110 \%\) of the current number of foxes.
• At time \(t = 0\), the number of foxes is 20 and the number of rabbits is 80 .

15

1. 1. Construct a mathematical model for the number of rabbits.
      [0pt] [9 marks]
      15
2. (ii) Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time.
   [0pt] [1 mark] 15
3. Suggest one way in which the model that you have used for the number of rabbits could be refined.
   [0pt] [1 mark]