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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]
AQA Further Paper 1 2022 June Q7
9 marks Standard +0.3
The matrix \(\mathbf{M}\) is defined as $$\mathbf{M} = \begin{bmatrix} 1 & 7 & -3 \\ 3 & 6 & k+1 \\ 1 & 3 & 2 \end{bmatrix}$$ where \(k\) is a constant.
    1. Given that \(\mathbf{M}\) is a non-singular matrix, find \(\mathbf{M}^{-1}\) in terms of \(k\) [5 marks]
    2. State any restrictions on the value of \(k\) [1 mark]
  2. Using your answer to part (a)(i), solve \begin{align} x + 7y - 3z &= 6
    3x + 6y + 6z &= 3
    x + 3y + 2z &= 1 \end{align} [3 marks]
AQA Further Paper 1 2022 June Q8
11 marks Standard +0.8
  1. The complex number \(w\) is such that $$\arg(w + 2i) = \tan^{-1}\frac{1}{2}$$ It is given that \(w = x + iy\), where \(x\) and \(y\) are real and \(x > 0\) Find an equation for \(y\) in terms of \(x\) [2 marks]
  2. The complex number \(z\) satisfies both $$-\frac{\pi}{2} \leq \arg(z + 2i) \leq \tan^{-1}\frac{1}{2} \quad \text{and} \quad |z - 2 + 3i| \leq 2$$ The region \(R\) is the locus of \(z\) Sketch the region \(R\) on the Argand diagram below. [4 marks] \includegraphics{figure_1}
  3. \(z_1\) is the point in \(R\) at which \(|z|\) is minimum.
    1. Calculate the exact value of \(|z_1|\) [3 marks]
    2. Express \(z_1\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2 marks]
AQA Further Paper 1 2022 June Q9
14 marks Challenging +1.8
Roberto is solving this mathematics problem:
The curve \(C_1\) has polar equation
\(r^2 = 9\sin 2\theta\)
for all possible values of \(\theta\)
Find the area enclosed by \(C_1\)
Roberto's solution is as follows:
\(A = \frac{1}{2}\int_{-\pi}^{\pi} 9\sin 2\theta \, d\theta\)
\(= \left[-\frac{9}{4}\cos 2\theta\right]_{-\pi}^{\pi}\)
\(= 0\)
  1. Sketch the curve \(C_1\) [2 marks]
  2. Explain what Roberto has done wrong. [2 marks]
  3. Find the area enclosed by \(C_1\) [2 marks]
  4. \(P\) and \(Q\) are distinct points on \(C_1\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\) [2 marks]
  5. The matrix \(\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) represents the transformation T T maps \(C_1\) onto a curve \(C_2\)
    1. T maps \(P\) onto the point \(P'\) Find the polar coordinates of \(P'\) [4 marks]
    2. Find the area enclosed by \(C_2\) Fully justify your answer. [2 marks]
AQA Further Paper 1 2022 June Q10
12 marks Challenging +1.8
In this question all measurements are in centimetres. A small, thin laser pen is set up with one end at \(A(7, 2, -3)\) and the other end at \(B(9, -3, -2)\) A laser beam travels from \(A\) to \(B\) and continues in a straight line towards a large thin sheet of glass. The sheet of glass lies within a plane \(\Pi_1\) which is modelled by the equation $$4x + py + 5z = 9$$ where \(p\) is an integer.
  1. The laser beam hits \(\Pi_1\) at an acute angle \(\alpha\), where \(\sin \alpha = \frac{\sqrt{15}}{75}\) Find the value of \(p\) [6 marks]
  2. A second large sheet of glass lies on the other side of \(\Pi_1\) This second sheet lies within a plane \(\Pi_2\) which is modelled by the equation $$4x + py + 5z = -5$$ Calculate the distance between the sheets of glass. [2 marks]
  3. The point \(A(7, 2, -3)\) is reflected in \(\Pi_1\) Find the coordinates of the image of \(A\) after reflection in \(\Pi_1\) [4 marks]
AQA Further Paper 1 2022 June Q11
19 marks Challenging +1.8
In this question use \(g\) as \(10\,\text{m}\,\text{s}^{-2}\) A smooth plane is inclined at \(30°\) to the horizontal. The fixed points \(A\) and \(B\) are 3.6 metres apart on the line of greatest slope of the plane, with \(A\) higher than \(B\) A particle \(P\) of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points \(A\) and \(B\) respectively. The particle \(P\) moves on a straight line that passes through \(A\) and \(B\) \includegraphics{figure_2} The natural length of the string \(AP\) is 1.4 metres. When the extension of the string \(AP\) is \(e_A\) metres, the tension in the string \(AP\) is \(7e_A\) newtons. The natural length of the string \(BP\) is 1 metre. When the extension of the string \(BP\) is \(e_B\) metres, the tension in the string \(BP\) is \(9e_B\) newtons. The particle \(P\) is held at the point between \(A\) and \(B\) which is 0.2 metres from its equilibrium position and lower than its equilibrium position. The particle \(P\) is then released from rest. At time \(t\) seconds after \(P\) is released, its displacement towards \(B\) from its equilibrium position is \(x\) metres.
  1. Show that during the subsequent motion the object satisfies the equation $$\ddot{x} + 50x = 0$$ Fully justify your answer. [5 marks]
  2. The experiment is repeated in a large tank of oil. During the motion the oil causes a resistive force of \(kv\) newtons to act on the particle, where \(v\,\text{m}\,\text{s}^{-1}\) is the speed of the particle. The oil causes critical damping to occur.
    1. Show that \(k = \frac{16\sqrt{2}}{5}\) [3 marks]
    2. Find \(x\) in terms of \(t\), giving your answer in exact form. [6 marks]
    3. Calculate the maximum speed of the particle. [5 marks]
AQA Further Paper 1 2022 June Q12
17 marks Challenging +1.2
The Argand diagram shows the solutions to the equation \(z^5 = 1\) \includegraphics{figure_3}
  1. Solve the equation $$z^5 = 1$$ giving your answers in the form \(z = \cos\theta + i\sin\theta\), where \(0 \leq \theta < 2\pi\) [2 marks]
  2. Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon. [2 marks]
  3. Show that if \(c = \cos\theta\), where \(z = \cos\theta + i\sin\theta\) is a solution to the equation \(z^5 = 1\), then \(c\) satisfies the equation $$16c^5 - 20c^3 + 5c - 1 = 0$$ [5 marks]
  4. The Argand diagram on page 22 is repeated below. \includegraphics{figure_4} Explain, with reference to the Argand diagram, why the expression $$16c^5 - 20c^3 + 5c - 1$$ has a repeated quadratic factor. [3 marks]
  5. \(O\) is the centre of a regular pentagon \(ABCDE\) such that \(OA = OB = OC = OD = OE = 1\) unit. The distance from \(O\) to \(AB\) is \(h\) By solving the equation \(16c^5 - 20c^3 + 5c - 1 = 0\), show that $$h = \frac{\sqrt{5} + 1}{4}$$ [5 marks]