Questions Further Paper 1 (256 questions)

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AQA Further Paper 1 Specimen Q10
10 marks Challenging +1.3
The curve, \(C\), has equation \(y = \frac{x}{\cosh x}\)
  1. Show that the \(x\)-coordinates of any stationary points of \(C\) satisfy the equation \(\tanh x = \frac{1}{x}\) [3 marks]
    1. Sketch the graphs of \(y = \tanh x\) and \(y = \frac{1}{x}\) on the axes below. [2 marks]
    2. Hence determine the number of stationary points of the curve \(C\). [1 mark]
  3. Show that \(\frac{d^2y}{dx^2} + y = 0\) at each of the stationary points of the curve \(C\). [4 marks]
AQA Further Paper 1 Specimen Q11
6 marks Standard +0.8
  1. Prove that \(\frac{\sinh \theta}{1 + \cosh \theta} + \frac{1 + \cosh \theta}{\sinh \theta} \equiv 2\coth \theta\) Explicitly state any hyperbolic identities that you use within your proof. [4 marks]
  2. Solve \(\frac{\sinh \theta}{1 + \cosh \theta} + \frac{1 + \cosh \theta}{\sinh \theta} = 4\) giving your answer in an exact form. [2 marks]
AQA Further Paper 1 Specimen Q12
3 marks Challenging +1.8
The function \(f(x) = \cosh(ix)\) 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. [3 marks]
AQA Further Paper 1 Specimen Q13
5 marks Standard +0.8
Given that \(\mathbf{M} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}\), prove that \(\mathbf{M}^n = \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}\) for all \(n \in \mathbb{N}\) [5 marks]
AQA Further Paper 1 Specimen Q14
12 marks Challenging +1.8
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 \(v\) ms\(^{-1}\), a force of magnitude \(|8Mx|\) acts on the particle directed towards \(O\). A resistive force, of magnitude \(4Mv\), also acts on \(P\).
  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. [8 marks]
  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. [4 marks]
AQA Further Paper 1 Specimen Q15
11 marks Challenging +1.8
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.
    1. Construct a mathematical model for the number of rabbits. [9 marks]
    2. Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time. [1 mark]
  2. Suggest one way in which the model that you have used for the number of rabbits could be refined. [1 mark]