AQA Further Paper 1 (Further Paper 1) 2021 June

Find $$\sum_{r=1}^{20}(r^2 - 2r)$$ Circle your answer. [1 mark] 2450 \quad 2660 \quad 5320 \quad 43680

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 \(10\)

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 \(y = 4x\)

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]

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]

1. Show that the equation $$(2z - z^*)^* = z^2$$ has exactly four solutions and state these solutions. [7 marks]
2. 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]

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}

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]

Use l'Hôpital's rule to show that $$\lim_{x \to \infty} (xe^{-x}) = 0$$ Fully justify your answer. [4 marks]

Evaluate the improper integral $$\int_0^8 \ln x \, \mathrm{d}x$$ showing the limiting process. [6 marks]

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]

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]

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]

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]

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]