AQA Further Paper 1 (Further Paper 1) Specimen

A student states that \(\int_0^{\pi/2} \frac{\cos x + \sin x}{\cos x - \sin x} \, dx\) 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. [2 marks]

\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) is a root of the equation \(p(z) = 0\)

1. Express \(p(z)\) as a product of quadratic factors with real coefficients. [5 marks]
2. Solve the equation \(p(z) = 0\). [1 mark]

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

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

1. Find the possible values of \(k\) [3 marks]
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. [5 marks]
3. Given further that the equations of the planes form a consistent system, find the solution of the system of equations. [3 marks]

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

1. Sketch the curve. [4 marks]
2. Hence sketch the graph of \(y = \left|\frac{5 - 4x}{1 + x}\right|\). [1 mark]

A line has Cartesian equations \(x - p = \frac{y + 2}{q} = 3 - z\) and a plane has equation \(\mathbf{r} \cdot \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix} = -3\)

1. In the case where the plane fully contains the line, find the values of \(p\) and \(q\). [3 marks]
2. In the case where the line intersects the plane at a single point, find the range of values of \(p\) and \(q\). [3 marks]
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 \(z = 0\)
   1. Find the value of \(q\). [4 marks]
   2. Find the value of \(p\). [3 marks]

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]
2. 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]

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]

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]

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]

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]

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. 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]