AQA FP2 (Further Pure Mathematics 2) 2016 June

1. Given that \(f(r) = \frac{1}{4r-1}\), show that $$f(r) - f(r+1) = \frac{A}{(4r-1)(4r+3)}$$ where \(A\) is an integer. [2 marks]
2. Use the method of differences to find the value of \(\sum_{r=1}^{50} \frac{1}{(4r-1)(4r+3)}\), giving your answer as a fraction in its simplest form. [4 marks]

The cubic equation \(3z^3 + pz^2 + 17z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2\mathrm{i}\).

1. 1. Write down the value of another non-real root, \(\beta\), of this equation. [1 mark]
   2. Hence find the value of \(\alpha\beta\). [1 mark]
2. Find the value of the third root, \(\gamma\), of this equation. [3 marks]
3. Find the values of \(p\) and \(q\). [3 marks]

The arc of the curve with equation \(y = 4 - \ln(1-x^2)\) from \(x = 0\) to \(x = \frac{3}{4}\) has length \(s\).

1. Show that \(s = \int_0^{\frac{3}{4}} \frac{\sqrt{1+x^2}}{1-x^2} \, dx\). [4 marks]
2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer. [6 marks]

1. Given that \(y = \tan^{-1} \sqrt{3x}\), find \(\frac{dy}{dx}\), giving your answer in terms of \(x\). [2 marks]
2. Hence, or otherwise, show that \(\int_{\frac{1}{3}}^1 \frac{1}{(1+3x)\sqrt{x}} \, dx = \frac{\sqrt{3}\pi}{n}\), where \(n\) is an integer. [4 marks]

1. Find the modulus of the complex number \(-4\sqrt{3} + 4\mathrm{i}\), giving your answer as an integer. [2 marks]
2. The locus of points, \(L\), satisfies the equation \(|z + 4\sqrt{3} - 4\mathrm{i}| = 4\).
   1. Sketch the locus \(L\) on the Argand diagram below. [3 marks]
   2. The complex number \(w\) lies on \(L\) so that \(-\pi < \arg w \leq \pi\). Find the least possible value of \(\arg w\), giving your answer in terms of \(\pi\). [2 marks]
3. Solve the equation \(z^3 = -4\sqrt{3} + 4\mathrm{i}\), giving your answers in the form \(re^{\mathrm{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). [5 marks]

1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that $$x = \ln(y + \sqrt{y^2 + 1}).$$ [4 marks]
2. A curve has equation \(y = 6\cosh^2 x + 5\sinh x\).
   1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number. [5 marks]
   2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh^{-1} 2\) has area \(A\). Show that $$A = a\cosh^{-1} 2 + b\sqrt{3} + c$$ where \(a\), \(b\) and \(c\) are integers. [5 marks]

Given that \(p \geq -1\), prove by induction that, for all integers \(n \geq 1\), $$(1 + p)^n \geq 1 + np$$ [6 marks]

1. By applying de Moivre's theorem to \((\cos \theta + \mathrm{i} \sin \theta)^4\), where \(\cos \theta \neq 0\), show that $$(1 + \mathrm{i} \tan \theta)^4 + (1 - \mathrm{i} \tan \theta)^4 = \frac{2\cos 4\theta}{\cos^4 \theta}$$ [3 marks]
2. Hence show that \(z = \mathrm{i} \tan \frac{\pi}{8}\) satisfies the equation \((1 + z)^4 + (1 - z)^4 = 0\), and express the three other roots of this equation in the form \(\mathrm{i} \tan \phi\), where \(0 < \phi < \pi\). [2 marks]
3. Use the results from part (b) to find the values of:
   1. \(\tan^2 \frac{\pi}{8} \tan^2 \frac{3\pi}{8}\); [4 marks]
   2. \(\tan^2 \frac{\pi}{8} + \tan^2 \frac{3\pi}{8}\). [4 marks]