AQA Further Paper 1 (Further Paper 1) 2023 June

Find the number of solutions of the equation \(\tanh x = \cosh x\) Circle your answer. [1 mark] \(0 \quad 1 \quad 2 \quad 3\)

The diagram below shows a locus on an Argand diagram. \includegraphics{figure_2} Which of the equations below represents the locus shown above? Circle your answer. [1 mark] \(|z - 2 + 3\mathrm{i}| = 2 \quad |z + 2 - 3\mathrm{i}| = 2 \quad |z - 2 + 3\mathrm{i}| = 4 \quad |z + 2 - 3\mathrm{i}| = 4\)

The matrix \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) represents a transformation. Which one of the points below is an invariant point under this transformation? Circle your answer. [1 mark] \((1, 1) \quad (0, 2) \quad (3, 0) \quad (2, 1)\)

The solution of a second order differential equation is \(f(t)\) The differential equation models heavy damping. Which one of the statements below could be true? Tick \((\checkmark)\) one box. [1 mark] \(f(t) = 2\mathrm{e}^{-t} \cos(3t) + 5\mathrm{e}^{-t} \sin(3t) \quad \square\) \(f(t) = 3\mathrm{e}^{-t} + 4t\mathrm{e}^{-t} \quad \square\) \(f(t) = 7\mathrm{e}^{-t} + 2\mathrm{e}^{-2t} \quad \square\) \(f(t) = 8\mathrm{e}^{-t} \cos(3t - 0.1) \quad \square\)

The function f is defined by $$f(r) = 2^r(r - 2) \quad (r \in \mathbb{Z})$$

1. Show that $$f(r + 1) - f(r) = r2^r$$ [2 marks]
2. Use the method of differences to show that $$\sum_{r=1}^n r2^r = 2^{n+1}(n - 1) + 2$$ [4 marks]

The matrix M is given by $$\mathbf{M} = \frac{1}{10} \begin{pmatrix} a & a & -6 \\ 0 & 10 & 0 \\ 9 & 14 & -13 \end{pmatrix}$$ where \(a\) is a real number. The vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are eigenvectors of \(\mathbf{M}\) The corresponding eigenvalues are \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) respectively. It is given that \(\lambda_2 = 1\) and \(\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) and \(\mathbf{v}_3 = \begin{pmatrix} c \\ 0 \\ 1 \end{pmatrix}\), where \(c\) is an integer.

1. 1. Find the value of \(\lambda_1\) [2 marks]
   2. Find the value of \(a\) [2 marks]
2. Find the integer \(c\) and the value of \(\lambda_3\) [4 marks]
3. Find matrices \(\mathbf{U}\), \(\mathbf{D}\) and \(\mathbf{U}^{-1}\), such that \(\mathbf{D}\) is diagonal and \(\mathbf{M} = \mathbf{UDU}^{-1}\) [3 marks]

The function f is defined by $$f(x) = \left|\sin x + \frac{1}{2}\right| \quad (0 \leq x \leq 2\pi)$$ Find the set of values of \(x\) for which $$f(x) \geq \frac{1}{2}$$ Give your answer in set notation. [5 marks]

The function g is defined by $$g(x) = \mathrm{e}^{\sin x} \quad (0 \leq x \leq 2\pi)$$ The diagram below shows the graph of \(y = g(x)\) \includegraphics{figure_8}

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = g(x)\), giving your answers in exact form. [1 mark]
2. Use Simpson's rule with 3 ordinates to estimate $$\int_0^\pi g(x) \, \mathrm{d}x$$ giving your answer to two decimal places. [3 marks]
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b). [1 mark]

The position vectors of the points \(A\), \(B\) and \(C\) are $$\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k}$$ $$\mathbf{b} = -\mathbf{i} - 8\mathbf{j} + 2\mathbf{k}$$ $$\mathbf{c} = -2\mathbf{j}$$ respectively.

1. Find the area of the triangle \(ABC\) [4 marks]
2. The points \(A\), \(B\) and \(C\) all lie in the plane \(\Pi\) Find an equation of the plane \(\Pi\), in the form \(\mathbf{r} \cdot \mathbf{n} = d\) [2 marks]
3. The point \(P\) has position vector \(\mathbf{p} = \mathbf{i} + 4\mathbf{j} + 2\mathbf{k}\) Find the exact distance of \(P\) from \(\Pi\) [3 marks]

The matrix M is defined as $$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ -1 & -1 & -2 \\ 1 & 2 & c \end{pmatrix}$$ where \(c\) is a real number.

1. The linear transformation T is represented by the matrix \(\mathbf{M}\) Show that, for one particular value of \(c\), the image under T of every point lies in the plane $$x + 5y + 3z = 0$$ State the value of \(c\) for which this occurs. [3 marks]
2. It is given that M is a non-singular matrix.
   1. State any restrictions on the value of \(c\) [2 marks]
   2. Find \(\mathbf{M}^{-1}\) in terms of \(c\) [4 marks]
   3. Using your answer from part (b)(ii), solve $$2x - y + z = -3$$ $$-x - y - 2z = -6$$ $$x + 2y + 4z = 13$$ [3 marks]

The function f is defined by $$f(x) = 4x^3 - 8x^2 - 51x - 45 \quad (x \in \mathbb{R})$$

1. 1. Fully factorise \(f(x)\) [2 marks]
   2. Hence, solve the inequality \(f(x) < 0\) [2 marks]
2. The graph of \(y = f(x)\) is translated by the vector \(\begin{pmatrix} 7 \\ 0 \end{pmatrix}\) The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = g(x)\) Solve the inequality \(g(x) \leq 0\) [3 marks]

1. Starting from the identities for \(\sinh 2x\) and \(\cosh 2x\), prove the identity $$\tanh 2x = \frac{2 \tanh x}{1 + \tanh^2 x}$$ [2 marks]
2. 1. The function f is defined by $$f(x) = \tanh x \quad (x > 0)$$ State the range of f [1 mark]
   2. Use part (a) and part (b)(i) to prove that \(\tanh 2x > \tanh x\) if \(x > 0\) [3 marks]

Use l'Hôpital's rule to prove that $$\lim_{x \to \pi} \frac{x \sin 2x}{\cos\left(\frac{x}{2}\right)} = -4\pi$$ [5 marks]

The curve C has polar equation $$r = \frac{A}{5 + 3 \cos \theta} \quad (-\pi < \theta \leq \pi)$$

1. Show that \(r\) takes values in the range \(\frac{1}{k} \leq r \leq k\), where \(k\) is an integer. [2 marks]
2. Find the Cartesian equation of C in the form \(y^2 = f(x)\) [4 marks]
3. The ellipse E has equation $$y^2 + \frac{16x^2}{25} = 1$$ Find the transformation that maps the graph of E onto C [4 marks]

Find the general solution of the differential equation $$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 3\frac{\mathrm{d}y}{\mathrm{d}x} - 4y = \cos 2x + 5x$$ [9 marks]

1. Show that $$\int_{0.5}^4 \frac{1}{t} \ln t \, \mathrm{d}t = a(\ln 2)^2$$ where \(a\) is a rational number to be found. [4 marks]
2. A curve C is defined parametrically for \(t > 0\) by $$x = 2t \quad y = \frac{1}{2}t^2 - \ln t$$ The arc formed by the graph of C from \(t = 0.5\) to \(t = 4\) is rotated through \(2\pi\) radians about the \(x\)-axis to generate a surface with area \(S\) Find the exact value of \(S\), giving your answer in the form $$S = \pi\left(b + c \ln 2 + d(\ln 2)^2\right)$$ where \(b\), \(c\) and \(d\) are rational numbers to be found. [7 marks]