AQA FP2 (Further Pure Mathematics 2) 2013 January

1. Show that $$12 \cosh x - 4 \sinh x = 4\text{e}^x + 8\text{e}^{-x}$$ [2 marks]
2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\). [5 marks]

Two loci, \(L_1\) and \(L_2\), in an Argand diagram are given by $$L_1 : |z + 6 - 5\text{i}| = 4\sqrt{2}$$ $$L_2 : \arg(z + \text{i}) = \frac{3\pi}{4}$$ The point \(P\) represents the complex number \(-2 + \text{i}\).

1. Verify that the point \(P\) is a point of intersection of \(L_1\) and \(L_2\). [2 marks]
2. Sketch \(L_1\) and \(L_2\) on one Argand diagram. [6 marks]
3. The point \(Q\) is also a point of intersection of \(L_1\) and \(L_2\). Find the complex number that is represented by \(Q\). [2 marks]

1. Show that \(\frac{1}{5r-2} - \frac{1}{5r+3} = \frac{A}{(5r-2)(5r+3)}\), stating the value of the constant \(A\). [2 marks]
2. Hence use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(5r-2)(5r+3)} = \frac{n}{3(5n+3)}$$ [4 marks]
3. Find the value of $$\sum_{r=1}^{\infty} \frac{1}{(5r-2)(5r+3)}$$ [1 mark]

The roots of the equation $$z^3 - 5z^2 + kz - 4 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha\beta\gamma\). [2 marks]
   2. Hence find the value of \(\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2\). [2 marks]
2. The value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\) is \(-4\).
   1. Explain why \(\alpha\), \(\beta\) and \(\gamma\) cannot all be real. [1 mark]
   2. By considering \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\), find the possible values of \(k\). [4 marks]

1. Using the definition \(\tanh y = \frac{\text{e}^y - \text{e}^{-y}}{\text{e}^y + \text{e}^{-y}}\), show that, for \(|x| < 1\), $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$$ [3 marks]
2. Hence, or otherwise, show that \(\frac{\text{d}}{\text{d}x}(\tanh^{-1} x) = \frac{1}{1-x^2}\). [3 marks]
3. Use integration by parts to show that $$\int_{0}^{\frac{1}{4}} \tanh^{-1} x \, \text{d}x = \ln \left(\frac{3^m}{2^n}\right)$$ where \(m\) and \(n\) are positive integers. [5 marks]

A curve is defined parametrically by $$x = t^3 + 5, \quad y = 6t^2 - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).

1. Show that \(s = \int_{0}^{3} 3t\sqrt{t^2 + A} \, \text{d}t\), stating the value of the constant \(A\). [4 marks]
2. Hence show that \(s = 61\). [4 marks]

The polynomial \(\text{p}(n)\) is given by \(\text{p}(n) = (n-1)^3 + n^3 + (n+1)^3\).

1. 1. Show that \(\text{p}(k+1) - \text{p}(k)\), where \(k\) is a positive integer, is a multiple of 9. [3 marks]
   2. Prove by induction that \(\text{p}(n)\) is a multiple of 9 for all integers \(n \geqslant 1\). [4 marks]
2. Using the result from part (a)(ii), show that \(n(n^2 + 2)\) is a multiple of 3 for any positive integer \(n\). [2 marks]

1. Express \(-4 + 4\sqrt{3}\text{i}\) in the form \(r\text{e}^{\text{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). [3 marks]
2. 1. Solve the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\), giving your answers in the form \(r\text{e}^{\text{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). [4 marks]
   2. The roots of the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\) are represented by the points \(P\), \(Q\) and \(R\) on an Argand diagram. Find the area of the triangle \(PQR\), giving your answer in the form \(k\sqrt{3}\), where \(k\) is an integer. [3 marks]
3. By considering the roots of the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\), show that $$\cos\frac{2\pi}{9} + \cos\frac{4\pi}{9} + \cos\frac{8\pi}{9} = 0$$ [4 marks]