Questions Further Paper 2 (305 questions)

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AQA Further Paper 2 2020 June Q15
16 marks Challenging +1.2
The points \(A(7, 2, 8)\), \(B(7, -4, 0)\) and \(C(3, 3.2, 9.6)\) all lie in the plane \(\Pi\).
  1. Find a Cartesian equation of the plane \(\Pi\). [3 marks]
  2. The line \(L_1\) has equation \(\mathbf{r} = \begin{bmatrix} 5 \\ -0.4 \\ 4.8 \end{bmatrix} + \mu \begin{bmatrix} 15 \\ 3 \\ 4 \end{bmatrix}\)
    1. Show that \(L_1\) lies in the plane \(\Pi\). [2 marks]
    2. Show that every point on \(L_1\) is equidistant from \(B\) and \(C\). [4 marks]
  3. The line \(L_2\) lies in the plane \(\Pi\), and every point on \(L_2\) is equidistant from \(A\) and \(B\). Find an equation of the line \(L_2\) [4 marks]
  4. The points \(A\), \(B\) and \(C\) all lie on a circle \(G\). The point \(D\) is the centre of circle \(G\). Find the coordinates of \(D\). [3 marks]
AQA Further Paper 2 2023 June Q1
1 marks Easy -1.8
Given that \(y = \sin x + \sinh x\), find \(\frac{d^2y}{dx^2} + y\) Circle your answer. [1 mark] \(2\sin x\) \quad \(-2\sin x\) \quad \(2\sinh x\) \quad \(-2\sinh x\)
AQA Further Paper 2 2023 June Q2
1 marks Easy -1.8
Which one of the expressions below is not equal to zero? Circle your answer. [1 mark] \(\lim_{x \to \infty} (x^2e^{-x})\) \quad \(\lim_{x \to 0} (x^5 \ln x)\) \quad \(\lim_{x \to \infty} \left(\frac{e^x}{x^5}\right)\) \quad \(\lim_{x \to 0^+} (x^3e^x)\)
AQA Further Paper 2 2023 June Q3
1 marks Standard +0.3
The determinant \(A = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) Which one of the determinants below has a value which is not equal to the value of \(A\)? Tick (\(\checkmark\)) one box. [1 mark] \(\begin{vmatrix} 3 & 1 & 3 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 2 & 3 \\ 1 & 0 & 2 \\ 1 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 2 & 2 & 2 \\ 1 & 0 & 1 \\ 3 & 2 & 1 \end{vmatrix}\) \quad \(\square\) \(\begin{vmatrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 0 & 2 \end{vmatrix}\) \quad \(\square\)
AQA Further Paper 2 2023 June Q4
1 marks Easy -1.2
It is given that \(f(x) = \cosh^{-1}(x - 3)\) Which of the sets listed below is the greatest possible domain of the function \(f\)? Circle your answer. [1 mark] \(\{x : x \geq 4\}\) \quad \(\{x : x \geq 3\}\) \quad \(\{x : x \geq 1\}\) \quad \(\{x : x \geq 0\}\)
AQA Further Paper 2 2023 June Q5
5 marks Challenging +1.2
Josh and Zoe are solving the following mathematics problem: The curve \(C_1\) has equation $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ The matrix \(\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) maps \(C_1\) onto \(C_2\) Find the equations of the asymptotes of \(C_2\) Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\) Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\) Show that Zoe is correct. [5 marks]
AQA Further Paper 2 2023 June Q6
5 marks Standard +0.3
  1. Express \(-5 - 5\text{i}\) in the form \(re^{i\theta}\), where \(-\pi < \theta \leq \pi\) [2 marks]
  2. The point on an Argand diagram that represents \(-5 - 5\text{i}\) is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle. Give your answers in the form \(re^{i\theta}\), where \(-\pi < \theta \leq \pi\) [3 marks]
AQA Further Paper 2 2023 June Q7
3 marks Standard +0.8
Show that $$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
AQA Further Paper 2 2023 June Q8
6 marks Standard +0.3
\(\mathbf{A}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{A}^T\) is the transpose of \(\mathbf{A}\)
  1. Using the result $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$ show that $$(\mathbf{A}^{-1})^T = (\mathbf{A}^T)^{-1}$$ [3 marks]
  2. It is given that \(\mathbf{A} = \begin{pmatrix} 4 & 5 \\ -1 & k \end{pmatrix}\), where \(k\) is a real constant.
    1. Find \((\mathbf{A}^{-1})^T\), giving your answer in terms of \(k\) [2 marks]
    2. State the restriction on the possible values of \(k\) [1 mark]
AQA Further Paper 2 2023 June Q9
7 marks Challenging +1.2
The complex number \(z\) is such that $$z = \frac{1 + \text{i}}{1 - k\text{i}}$$ where \(k\) is a real number.
  1. Find the real part of \(z\) and the imaginary part of \(z\), giving your answers in terms of \(k\) [2 marks]
  2. In the case where \(k = \sqrt{3}\), use part (a) to show that $$\cos \frac{7\pi}{12} = \frac{\sqrt{2} - \sqrt{6}}{4}$$ [5 marks]
AQA Further Paper 2 2023 June Q10
8 marks Challenging +1.2
The region \(R\) on an Argand diagram satisfies both \(|z + 2\text{i}| \leq 3\) and \(-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}\)
  1. Sketch \(R\) on the Argand diagram below. [3 marks] \includegraphics{figure_10a}
  2. Find the maximum value of \(|z|\) in the region \(R\), giving your answer in exact form. [5 marks]
AQA Further Paper 2 2023 June Q11
9 marks Standard +0.8
The line \(l_1\) passes through the points \(A(6, 2, 7)\) and \(B(4, -3, 7)\)
  1. Find a Cartesian equation of \(l_1\) [2 marks]
  2. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 8 \\ 9 \\ c \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\) where \(c\) is a constant.
    1. Explain how you know that the lines \(l_1\) and \(l_2\) are not perpendicular. [2 marks]
    2. The lines \(l_1\) and \(l_2\) both lie in the same plane. Find the value of \(c\) [5 marks]
AQA Further Paper 2 2023 June Q12
6 marks Standard +0.3
The function \(f\) is defined by $$f(n) = 3^{3n+1} + 2^{3n+4} \quad (n \in \mathbb{Z}^+)$$ Prove by induction that \(f(n)\) is divisible by 19 for \(n \geq 1\) [6 marks]
AQA Further Paper 2 2023 June Q13
11 marks Challenging +1.8
The quadratic equation \(z^2 - 5z + 8 = 0\) has roots \(\alpha\) and \(\beta\)
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\) [2 marks]
  2. Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha^4 + \beta^4 = -47\) [4 marks]
  3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha^3 + \beta\) and \(\beta^3 + \alpha\) [5 marks]
AQA Further Paper 2 2023 June Q14
10 marks Challenging +1.3
The function \(f\) is defined by $$f(x) = \frac{1}{4x^2 + 16x + 19} \quad (x \in \mathbb{R})$$
  1. Show, without using calculus, that the graph of \(y = f(x)\) has a stationary point at \(\left(-2, \frac{1}{3}\right)\) [3 marks]
  2. Show that \(\int_{-2}^{-\frac{1}{2}} f(x) \, dx = \frac{\pi\sqrt{3}}{18}\) [5 marks]
  3. Find the value of \(\int_{-2}^{\infty} f(x) \, dx\) Fully justify your answer. [2 marks]
AQA Further Paper 2 2023 June Q15
10 marks Challenging +1.2
  1. Given that \(z = \cos \theta + \text{i} \sin \theta\), use de Moivre's theorem to show that $$z^n - z^{-n} = 2\text{i} \sin n\theta$$ [2 marks]
  2. The series \(S\) is defined as $$S = \sin \theta + \sin 3\theta + \ldots + \sin(2n - 1)\theta$$ Use part (a) to express \(S\) in the form $$S = \frac{1}{2\text{i}}(G_1) - \frac{1}{2\text{i}}(G_2)$$ where each of \(G_1\) and \(G_2\) is a geometric series. [3 marks]
  3. Hence, show that $$S = \frac{\sin^2(n\theta)}{\sin \theta}$$ [5 marks]
AQA Further Paper 2 2023 June Q16
16 marks Hard +2.3
A bungee jumper of mass \(m\) kg is attached to an elastic rope. The other end of the rope is attached to a fixed point. The bungee jumper falls vertically from the fixed point. At time \(t\) seconds after the rope first becomes taut, the extension of the rope is \(x\) metres and the speed of the bungee jumper is \(v\) m s\(^{-1}\)
  1. A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are • the weight of the bungee jumper • a tension in the rope of magnitude \(kx\) newtons • an air resistance force of magnitude \(Rv\) newtons where \(k\) and \(R\) are constants such that \(4km > R^2\)
    1. Show that this model gives the result $$x = e^{-\frac{Rt}{2m}} \left( A \cos \frac{\sqrt{4km - R^2}}{2m} t + B \sin \frac{\sqrt{4km - R^2}}{2m} t \right) + \frac{mg}{k}$$ where \(A\) and \(B\) are constants, and \(g\) m s\(^{-2}\) is the acceleration due to gravity. You do not need to find the value of \(A\) or the value of \(B\) [6 marks]
    2. It is also given that: \(k = 16\) \(R = 20\) \(m = 62.5\) \(g = 9.8\) m s\(^{-2}\) and that the speed of the bungee jumper when the rope becomes taut is 14 m s\(^{-1}\) Show that, to the nearest integer, \(A = -38\) and \(B = 16\) [6 marks]
  2. A second, simpler model assumes that the air resistance is zero. The values of \(k\), \(m\) and \(g\) remain the same. Find an expression for \(x\) in terms of \(t\) according to this simpler model, giving the values of all constants to two significant figures. [4 marks]
AQA Further Paper 2 2024 June Q1
1 marks Easy -1.8
It is given that $$\begin{bmatrix} 2 \\ 1 \\ 9 \end{bmatrix} \times \begin{bmatrix} 5 \\ \lambda \\ -6 \end{bmatrix} = 0$$ where \(\lambda\) is a constant. Find the value of \(\lambda\) Circle your answer. [1 mark] \(-28\) \quad\quad \(-8\) \quad\quad \(8\) \quad\quad \(28\)
AQA Further Paper 2 2024 June Q2
1 marks Moderate -0.8
The movement of a particle is described by the simple harmonic equation $$\ddot{x} = -25x$$ where \(x\) metres is the displacement of the particle at time \(t\) seconds, and \(\ddot{x}\) m s\(^{-2}\) is the acceleration of the particle. The maximum displacement of the particle is 9 metres. Find the maximum speed of the particle. Circle your answer. [1 mark] \(15\) m s\(^{-1}\) \quad\quad \(45\) m s\(^{-1}\) \quad\quad \(75\) m s\(^{-1}\) \quad\quad \(135\) m s\(^{-1}\)
AQA Further Paper 2 2024 June Q3
1 marks Easy -1.2
The function g is defined by $$g(x) = \text{sech } x \quad\quad (x \in \mathbb{R})$$ Which one of the following is the range of g? Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < g(x) \leq -1\) \quad \(\square\) \(-1 \leq g(x) < 0\) \quad \(\square\) \(0 < g(x) \leq 1\) \quad \(\square\) \(1 \leq g(x) \leq \infty\) \quad \(\square\)
AQA Further Paper 2 2024 June Q4
1 marks Moderate -0.8
The function f is a quartic function with real coefficients. The complex number \(5i\) is a root of the equation \(f(x) = 0\) Which one of the following must be a factor of \(f(x)\)? Circle your answer. [1 mark] \((x^2 - 25)\) \quad\quad \((x^2 - 5)\) \quad\quad \((x^2 + 5)\) \quad\quad \((x^2 + 25)\)
AQA Further Paper 2 2024 June Q5
3 marks Standard +0.3
The first four terms of the series \(S\) can be written as $$S = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + ...$$
  1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) [1 mark]
  2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac{1}{3}n(n + 1)(n + 2)$$ [2 marks]
AQA Further Paper 2 2024 June Q6
3 marks Moderate -0.3
The cubic equation $$x^3 + 5x^2 - 4x + 2 = 0$$ has roots \(\alpha\), \(\beta\) and \(\gamma\) Find a cubic equation, with integer coefficients, whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\) [3 marks]
AQA Further Paper 2 2024 June Q7
4 marks Standard +0.8
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows. $$\mathbf{A} = \begin{bmatrix} p - 2 & p - 1 \\ 0 & 1 \end{bmatrix} \quad\quad \mathbf{B} = \begin{bmatrix} 1 & 2p - 1 \\ 0 & 4 - p \end{bmatrix}$$ Find the values of \(p\) such that \(\mathbf{A}\) and \(\mathbf{B}\) are commutative under matrix multiplication. Fully justify your answer. [4 marks]
AQA Further Paper 2 2024 June Q8
4 marks Standard +0.8
The vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are such that \(\mathbf{a} \times \mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(\mathbf{a} \times \mathbf{c} = \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}\) Work out \((\mathbf{a} - 4\mathbf{b} + 3\mathbf{c}) \times (2\mathbf{a})\) [4 marks]