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AQA Further Paper 1 Specimen Q8
5 marks Standard +0.3
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]
AQA Further Paper 1 Specimen Q9
13 marks Challenging +1.3
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]
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]
AQA Further Paper 2 2019 June Q1
1 marks Easy -1.8
Given that \(z\) is a complex number, and that \(z^*\) is the complex conjugate of \(z\), which of the following statements is not always true? Circle your answer. [1 mark] \((z^*)^* = z\) \quad\quad \(zz^* = |z|^2\) \quad\quad \((-z)^* = -(z^*)\) \quad\quad \(z - z^* = z^* - z\)
AQA Further Paper 2 2019 June Q2
1 marks Moderate -0.8
Which of the straight lines given below is an asymptote to the curve $$y = \frac{ax^2}{x-1}$$ where \(a\) is a non-zero constant? Circle your answer. [1 mark] \(y = ax + a\) \quad\quad \(y = ax\) \quad\quad \(y = ax - a\) \quad\quad \(y = a\)
AQA Further Paper 2 2019 June Q3
1 marks Moderate -0.8
The set \(A\) is defined by \(A = \{x : -\sqrt{2} < x < 0\} \cup \{x : 0 < x < \sqrt{2}\}\) Which of the inequalities given below has \(A\) as its solution? Circle your answer. [1 mark] \(|x^2 - 1| > 1\) \quad\quad \(|x^2 - 1| \geq 1\) \quad\quad \(|x^2 - 1| < 1\) \quad\quad \(|x^2 - 1| \leq 1\)
AQA Further Paper 2 2019 June Q4
3 marks Standard +0.3
The positive integer \(k\) is such that $$\sum_{r=1}^{k} (3r - k) = 90$$ Find the value of \(k\). [3 marks]
AQA Further Paper 2 2019 June Q5
4 marks Standard +0.8
A curve has equation \(y = \cosh x\) Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to $$\sinh b - \sinh a$$ [4 marks]
AQA Further Paper 2 2019 June Q6
6 marks Challenging +1.8
A circle \(C\) in the complex plane has equation \(|z - 2 - 5\mathrm{i}| = a\) The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(\arg(z_1) = \frac{\pi}{4}\) Prove that \(a = \frac{3\sqrt{2}}{2}\) [6 marks]
AQA Further Paper 2 2019 June Q7
6 marks Standard +0.3
The points \(A\), \(B\) and \(C\) have coordinates \(A(4, 5, 2)\), \(B(-3, 2, -4)\) and \(C(2, 6, 1)\)
  1. Use a vector product to show that the area of triangle \(ABC\) is \(\frac{5\sqrt{11}}{2}\) [4 marks]
  2. The points \(A\), \(B\) and \(C\) lie in a plane. Find a vector equation of the plane in the form \(\mathbf{r} \cdot \mathbf{n} = k\) [1 mark]
  3. Hence find the exact distance of the plane from the origin. [1 mark]
AQA Further Paper 2 2019 June Q8
9 marks Challenging +1.8
A parabola \(P_1\) has equation \(y^2 = 4ax\) where \(a > 0\) \(P_1\) is translated by the vector \(\begin{bmatrix} b \\ 0 \end{bmatrix}\), where \(b > 0\), to give the parabola \(P_2\)
  1. The line \(y = mx\) is a tangent to \(P_2\) Prove that \(m = \pm\sqrt{\frac{a}{b}}\) Solutions using differentiation will be given no marks. [4 marks]
  2. The line \(y = \sqrt{\frac{a}{b}} x\) meets \(P_2\) at the point \(D\). The finite region \(R\) is bounded by the \(x\)-axis, \(P_2\) and a line through \(D\) perpendicular to the \(x\)-axis. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis to form a solid. Find, in terms of \(a\) and \(b\), the volume of this solid. Fully justify your answer. [5 marks]
AQA Further Paper 2 2019 June Q9
13 marks Challenging +1.8
  1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 5 & 5 \\ -3 & 13 \\ 5 & 10 \end{bmatrix}$$ [5 marks]
  2. Find matrices \(\mathbf{U}\) and \(\mathbf{D}\) such that \(\mathbf{D}\) is a diagonal matrix and \(\mathbf{M} = \mathbf{U}\mathbf{D}\mathbf{U}^{-1}\) [2 marks]
  3. Given that \(\mathbf{M}^n \to \mathbf{L}\) as \(n \to \infty\), find the matrix \(\mathbf{L}\). [4 marks]
  4. The transformation represented by \(\mathbf{L}\) maps all points onto a line. Find the equation of this line. [2 marks]
AQA Further Paper 2 2019 June Q10
7 marks Standard +0.3
Prove by induction that \(f(n) = n^3 + 3n^2 + 8n\) is divisible by 6 for all integers \(n \geq 1\) [7 marks]
AQA Further Paper 2 2019 June Q11
8 marks Challenging +1.2
The line \(L_1\) has equation $$\frac{x-2}{3} = \frac{y+4}{8} = \frac{4z-5}{5}$$ The line \(L_2\) has equation $$\left(\mathbf{r} - \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}\right) \times \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \mathbf{0}$$ Find the shortest distance between the two lines, giving your answer to three significant figures. [8 marks]
AQA Further Paper 2 2019 June Q12
5 marks Challenging +1.2
Abel and Bonnie are trying to solve this mathematical problem: \(z = 2 - 3\mathrm{i}\) is a root of the equation \(2z^3 + mz^2 + pz + 91 = 0\) Find the value of \(m\) and the value of \(p\). Abel says he has solved the problem. Bonnie says there is not enough information to solve the problem.
  1. Abel's solution begins as follows: Since \(z = 2 - 3\mathrm{i}\) is a root of the equation, \(z = 2 + 3\mathrm{i}\) is another root. State one extra piece of information about \(m\) and \(p\) which could be added to the problem to make the beginning of Abel's solution correct. [1 mark]
  2. Prove that Bonnie is right. [4 marks]
AQA Further Paper 2 2019 June Q13
10 marks Standard +0.8
  1. Explain why \(\int_3^{\infty} x^2 e^{-2x} \, dx\) is an improper integral. [1 mark]
  2. Evaluate \(\int_3^{\infty} x^2 e^{-2x} \, dx\) Show the limiting process. [9 marks]
AQA Further Paper 2 2019 June Q14
12 marks Challenging +1.8
Let $$S_n = \sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}$$ where \(n \geq 1\)
  1. Use the method of differences to show that $$S_n = \frac{5n^2 + an}{12(n+b)(n+c)}$$ where \(a\), \(b\) and \(c\) are integers. [6 marks]
  2. Show that, for any number \(k\) greater than \(\frac{12}{5}\), if the difference between \(\frac{5}{12}\) and \(S_n\) is less than \(\frac{1}{k}\), then $$n > \frac{k-5+\sqrt{k^2+1}}{2}$$ [6 marks]
AQA Further Paper 2 2019 June Q15
14 marks Challenging +1.8
\includegraphics{figure_15} Two tanks, A and B, each have a capacity of 800 litres. At time \(t = 0\) both tanks are full of pure water. When \(t > 0\), water flows in the following ways: • Water with a salt concentration of \(\mu\) grams per litre flows into tank A at a constant rate • Water flows from tank A to tank B at a rate of 16 litres per minute • Water flows from tank B to tank A at a rate of \(r\) litres per minute • Water flows out of tank B through a waste pipe • The amount of water in each tank remains at 800 litres. At time \(t\) minutes (\(t \geq 0\)) there are \(x\) grams of salt in tank A and \(y\) grams of salt in tank B. This system is represented by the coupled differential equations \begin{align} \frac{dx}{dt} &= 36 - 0.02x + 0.005y \tag{1}
\frac{dy}{dt} &= 0.02x - 0.02y \tag{2} \end{align}
  1. Find the value of \(r\). [2 marks]
  2. Show that \(\mu = 3\) [3 marks]
  3. Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\). [9 marks]
AQA Further Paper 2 2020 June Q1
1 marks Moderate -0.8
Three of the four expressions below are equivalent to each other. Which of the four expressions is not equivalent to any of the others? Circle your answer. [1 mark] \(\mathbf{a} \times (\mathbf{a} + \mathbf{b})\) \quad \((\mathbf{a} + \mathbf{b}) \times \mathbf{b}\) \quad \((\mathbf{a} - \mathbf{b}) \times \mathbf{b}\) \quad \(\mathbf{a} \times (\mathbf{a} - \mathbf{b})\)
AQA Further Paper 2 2020 June Q2
1 marks Standard +0.3
Given that \(\arg(a + bi) = \varphi\), where \(a\) and \(b\) are positive real numbers and \(0 < \varphi < \frac{\pi}{2}\), three of the following four statements are correct. Which statement is not correct? Tick \((\checkmark)\) one box. [1 mark] \(\arg(-a - bi) = \pi - \varphi\) \(\arg(a - bi) = -\varphi\) \(\arg(b + ai) = \frac{\pi}{2} - \varphi\) \(\arg(b - ai) = \varphi - \frac{\pi}{2}\)