Questions — AQA Further Paper 1 (105 questions)

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AQA Further Paper 1 2019 June Q12
8 marks Challenging +1.8
Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} 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. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]
AQA Further Paper 1 2019 June Q13
14 marks Challenging +1.8
The equation \(z^3 + kz^2 + 9 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
    1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = k^2$$ [3 marks]
    2. Show that $$\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -18k$$ [4 marks]
  2. The equation \(9z^3 - 40z^2 + rz + s = 0\) has roots \(\alpha\beta + \gamma\), \(\beta\gamma + \alpha\) and \(\gamma\alpha + \beta\).
    1. Show that $$k = -\frac{40}{9}$$ [1 mark]
    2. Without calculating the values of \(\alpha\), \(\beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer. [6 marks]
AQA Further Paper 1 2019 June Q14
11 marks Challenging +1.8
In this question use \(g = 10 \text{ m s}^{-2}\) A light spring is attached to the base of a long tube and has a mass \(m\) attached to the other end, as shown in the diagram. The tube is filled with oil. When the compression of the spring is \(c\) metres, the thrust in the spring is \(9mc\) newtons. \includegraphics{figure_14} The mass is held at rest in a position where the compression of the spring is \(\frac{20}{9}\) metres. The mass is then released from rest. During the subsequent motion the oil causes a resistive force of \(6mv\) newtons to act on the mass, where \(v \text{ m s}^{-1}\) is the speed of the mass. At time \(t\) seconds after the mass is released, the displacement of the mass above its starting position is \(x\) metres.
  1. Find \(x\) in terms of \(t\). [10 marks]
  2. State, giving a reason, the type of damping which occurs. [1 mark]
AQA Further Paper 1 2019 June Q15
11 marks Challenging +1.8
The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \((r, \theta)\) where \(0 \leq \theta \leq \frac{\pi}{2}\) The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole. \(TPQ\) is the tangent to the curve at \(P\). \includegraphics{figure_15}
  1. Show that the gradient of \(TPQ\) is equal to $$\frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$ [4 marks]
  2. The curve has polar equation $$r = e^{(\cot b)\theta}$$ where \(b\) is a constant such that \(0 < b < \frac{\pi}{2}\) Use the result of part (a) to show that the angle between the line \(OP\) and the tangent \(TPQ\) does not depend on \(\theta\). [7 marks]
AQA Further Paper 1 2021 June Q1
1 marks Easy -1.2
Find $$\sum_{r=1}^{20}(r^2 - 2r)$$ Circle your answer. [1 mark] 2450 \quad 2660 \quad 5320 \quad 43680
AQA Further Paper 1 2021 June Q2
1 marks Moderate -0.8
Given that \(z = 1 - 3\mathrm{i}\) is one root of the equation \(z^2 + pz + r = 0\), where \(p\) and \(r\) are real, find the value of \(r\). Circle your answer. [1 mark] \(-8\) \quad \(-2\) \quad \(6\) \quad \(10\)
AQA Further Paper 1 2021 June Q3
1 marks Moderate -0.5
The curve C has polar equation $$r^2 \sin 2\theta = 4$$ Find a Cartesian equation for C. Circle your answer. [1 mark] \(y = 2x\) \quad \(y = \frac{x}{2}\) \quad \(y = \frac{2}{x}\) \quad \(y = 4x\)
AQA Further Paper 1 2021 June Q4
5 marks Challenging +1.2
Show that the solutions to the equation $$3\tanh^2 x - 2\operatorname{sech} x = 2$$ can be expressed in the form $$x = \pm \ln(a + \sqrt{b})$$ where \(a\) and \(b\) are integers to be found. You may use without proof the result \(\cosh^{-1} y = \ln(y + \sqrt{y^2 - 1})\) [5 marks]
AQA Further Paper 1 2021 June Q5
5 marks Standard +0.8
The matrix M is defined by \(\mathbf{M} = \begin{pmatrix} 3 & 2 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) for all integers \(n \geq 1\) [5 marks]
AQA Further Paper 1 2021 June Q6
10 marks Challenging +1.8
  1. Show that the equation $$(2z - z^*)^* = z^2$$ has exactly four solutions and state these solutions. [7 marks]
    1. Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry. [2 marks] \includegraphics{figure_6b}
    2. Show that the area of this quadrilateral is \(\frac{\sqrt{15}}{2}\) square units. [1 mark]
AQA Further Paper 1 2021 June Q7
7 marks Standard +0.3
The diagram below shows the graph of \(y = \mathrm{f}(x)\) (\(-4 \leq x \leq 4\)) The graph meets the \(x\)-axis at \(x = 1\) and \(x = 3\) The graph meets the \(y\)-axis at \(y = 2\) \includegraphics{figure_7}
  1. Sketch the graph of \(y = |\mathrm{f}(x)|\) on the axes below. Show any axis intercepts. [2 marks] \includegraphics{figure_7a}
  2. Sketch the graph of \(y = \frac{1}{\mathrm{f}(x)}\) on the axes below. Show any axis intercepts and asymptotes. [3 marks] \includegraphics{figure_7b}
  3. Sketch the graph of \(y = \mathrm{f}(|x|)\) on the axes below. Show any axis intercepts. [2 marks] \includegraphics{figure_7c}
AQA Further Paper 1 2021 June Q8
6 marks Challenging +1.2
A particle of mass 4 kg moves horizontally in a straight line. At time \(t\) seconds the velocity of the particle is \(v\) m s\(^{-1}\) The following horizontal forces act on the particle: • a constant driving force of magnitude 1.8 newtons • another driving force of magnitude \(30\sqrt{t}\) newtons • a resistive force of magnitude \(0.08v^2\) newtons When \(t = 70\), \(v = 54\) Use Euler's method with a step length of 0.5 to estimate the velocity of the particle after 71 seconds. Give your answer to four significant figures. [6 marks]
AQA Further Paper 1 2021 June Q9
4 marks Moderate -0.5
Use l'Hôpital's rule to show that $$\lim_{x \to \infty} (xe^{-x}) = 0$$ Fully justify your answer. [4 marks]
AQA Further Paper 1 2021 June Q10
6 marks Challenging +1.2
Evaluate the improper integral $$\int_0^8 \ln x \, \mathrm{d}x$$ showing the limiting process. [6 marks]
AQA Further Paper 1 2021 June Q11
12 marks Standard +0.8
The line \(L_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}\) The line \(L_2\) has equation \(\mathbf{r} = \begin{pmatrix} 6 \\ 4 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix}\)
  1. Find the acute angle between the lines \(L_1\) and \(L_2\), giving your answer to the nearest 0.1° [3 marks]
  2. The lines \(L_1\) and \(L_2\) lie in the plane \(\Pi_1\)
    1. Find the equation of \(\Pi_1\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = d\) [4 marks]
    2. Hence find the shortest distance of the plane \(\Pi_1\) from the origin. [1 mark]
  3. The points \(A(4, -1, -1)\), \(B(1, 5, -7)\) and \(C(3, 4, -8)\) lie in the plane \(\Pi_2\) Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), giving your answer to the nearest 0.1° [4 marks]
AQA Further Paper 1 2021 June Q12
14 marks Standard +0.8
The matrix \(\mathbf{A} = \begin{pmatrix} 1 & 5 & 3 \\ 4 & -2 & p \\ 8 & 5 & -11 \end{pmatrix}\), where \(p\) is a constant.
  1. Given that A is a non-singular matrix, find \(\mathbf{A}^{-1}\) in terms of \(p\). State any restrictions on the value of \(p\). [6 marks]
  2. The equations below represent three planes. \(x + 5y + 3z = 5\) \(4x - 2y + pz = 24\) \(8x + 5y - 11z = -30\)
    1. Find, in terms of \(p\), the coordinates of the point of intersection of the three planes. [4 marks]
    2. In the case where \(p = 2\), show that the planes are mutually perpendicular. [4 marks]
AQA Further Paper 1 2021 June Q13
3 marks Standard +0.8
The transformation S is represented by the matrix \(\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}\) The transformation T is a translation by the vector \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\) Kamla transforms the graphs of various functions by applying first S, then T. Leo says that, for some graphs, Kamla would get a different result if she applied first T, then S. Kamla disagrees. State who is correct. Fully justify your answer. [3 marks]
AQA Further Paper 1 2021 June Q14
12 marks Challenging +1.8
The hyperbola \(H\) has equation \(y^2 - x^2 = 16\) The circle \(C\) has equation \(x^2 + y^2 = 32\) The diagram below shows part of the graph of \(H\) and part of the graph of \(C\). \includegraphics{figure_14} Show that the shaded region in the first quadrant enclosed by \(H\), \(C\), the \(x\)-axis and the \(y\)-axis has area $$\frac{16\pi}{3} + 8\ln\left(\frac{\sqrt{2} + \sqrt{6}}{2}\right)$$ [12 marks]
AQA Further Paper 1 2021 June Q15
13 marks Challenging +1.8
In this question use \(g = 9.8\) m s\(^{-2}\) A particle \(P\) of mass \(m\) is attached to two light elastic strings, \(AP\) and \(BP\). The other ends of the strings, \(A\) and \(B\), are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container. The coefficient of friction between \(P\) and the surface is 0.68 • When the extension of string \(AP\) is \(e_A\) metres, the tension in \(AP\) is \(24me_A\) • When the extension of string \(BP\) is \(e_B\) metres, the tension in \(BP\) is \(10me_B\) • The natural length of string \(AP\) is 1 metre • The natural length of string \(BP\) is 1.3 metres \includegraphics{figure_15}
  1. Show that when \(AP = 1.5\) metres, the tension in \(AP\) is equal to the tension in \(BP\). [1 mark]
  2. \(P\) is held at the point between \(A\) and \(B\) where \(AP = 1.9\) metres, and then released from rest. At time \(t\) seconds after \(P\) is released, \(AP = (1.5 + x)\) metres. \includegraphics{figure_15b} Show that when \(P\) is moving towards \(A\), $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 34x = 6.664$$ [3 marks]
  3. The container is then filled with oil, and \(P\) is again released from rest at the point between \(A\) and \(B\) where \(AP = 1.9\) metres. At time \(t\) seconds after \(P\) is released, the oil causes a resistive force of magnitude \(10mv\) newtons to act on the particle, where \(v\) m s\(^{-1}\) is the speed of the particle. Find \(x\) in terms of \(t\) when \(P\) is moving towards \(A\). [9 marks]
AQA Further Paper 1 2022 June Q1
1 marks Moderate -0.8
The displacement of a particle from its equilibrium position is \(x\) metres at time \(t\) seconds. The motion of the particle obeys the differential equation $$\frac{d^2x}{dt^2} = -9x$$ Calculate the period of its motion in seconds. Circle your answer. [1 mark] \(\frac{\pi}{9}\) \(\quad\) \(\frac{2\pi}{9}\) \(\quad\) \(\frac{\pi}{3}\) \(\quad\) \(\frac{2\pi}{3}\)
AQA Further Paper 1 2022 June Q2
1 marks Moderate -0.8
Simplify $$\frac{\cos\left(\frac{6\pi}{13}\right) + i\sin\left(\frac{6\pi}{13}\right)}{\cos\left(\frac{2\pi}{13}\right) - i\sin\left(\frac{2\pi}{13}\right)}$$ Tick (\(\checkmark\)) one box. [1 mark] \(\cos\left(\frac{8\pi}{13}\right) + i\sin\left(\frac{8\pi}{13}\right)\) \(\square\) \(\cos\left(\frac{8\pi}{13}\right) - i\sin\left(\frac{8\pi}{13}\right)\) \(\square\) \(\cos\left(\frac{4\pi}{13}\right) + i\sin\left(\frac{4\pi}{13}\right)\) \(\square\) \(\cos\left(\frac{4\pi}{13}\right) - i\sin\left(\frac{4\pi}{13}\right)\) \(\square\)
AQA Further Paper 1 2022 June Q3
1 marks Easy -1.2
Given that \(y = \operatorname{sech}x\), find \(\frac{dy}{dx}\) Tick (\(\checkmark\)) one box. [1 mark] \(\operatorname{sech}x\tanh x\) \(\square\) \(-\operatorname{sech}x\tanh x\) \(\square\) \(\operatorname{cosech}x\coth x\) \(\square\) \(-\operatorname{cosech}x\coth x\) \(\square\)
AQA Further Paper 1 2022 June Q4
1 marks Moderate -0.5
The vector \(\mathbf{v}\) is an eigenvector of the matrix \(\mathbf{N}\) with corresponding eigenvalue 4 The vector \(\mathbf{v}\) is also an eigenvector of the matrix \(\mathbf{M}\) with corresponding eigenvalue 3 Given that $$\mathbf{N}\mathbf{M}^2\mathbf{v} = \lambda\mathbf{v}$$ find the value of \(\lambda\) Circle your answer. [1 mark] 10 \(\quad\) 24 \(\quad\) 36 \(\quad\) 144
AQA Further Paper 1 2022 June Q5
6 marks Standard +0.8
It is given that \(z = -\frac{3}{2} + i\frac{\sqrt{11}}{2}\) is a root of the equation $$z^4 - 3z^3 - 5z^2 + kz + 40 = 0$$ where \(k\) is a real number.
  1. Find the other three roots. [5 marks]
  2. Given that \(x \in \mathbb{R}\), solve $$x^4 - 3x^3 - 5x^2 + kx + 40 < 0$$ [1 mark]
AQA Further Paper 1 2022 June Q6
8 marks Challenging +1.3
  1. Given that \(|x| < 1\), prove that $$\tanh^{-1}x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$$ [4 marks]
  2. Solve the equation $$20\operatorname{sech}^2x - 11\tanh x = 16$$ Give your answer in logarithmic form. [4 marks]