AQA FP2 (Further Pure Mathematics 2) 2011 June

1. Draw on the same Argand diagram:
   1. the locus of points for which $$|z - 2 - 5i| = 5$$ [3 marks]
   2. the locus of points for which $$\arg(z + 2i) = \frac{\pi}{4}$$ [3 marks]
2. Indicate on your diagram the set of points satisfying both $$|z - 2 - 5i| \leqslant 5$$ and $$\arg(z + 2i) = \frac{\pi}{4}$$ [2 marks]

1. Use the definitions of \(\cosh \theta\) and \(\sinh \theta\) in terms of \(e^\theta\) to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh(x - y)$$ [4 marks]
2. It is given that \(x\) satisfies the equation $$\cosh(x - \ln 2) = \sinh x$$
   1. Show that \(\tanh x = \frac{5}{4}\). [4 marks]
   2. Express \(x\) in the form \(\frac{1}{2} \ln a\). [2 marks]

1. Show that $$(r + 1)! - (r - 1)! = (r^2 + r - 1)(r - 1)!$$ [2 marks]
2. Hence show that $$\sum_{r=1}^{n} (r^2 + r - 1)(r - 1)! = (n + 2)n! - 2$$ [4 marks]

The cubic equation $$z^3 - 2z^2 + k = 0 \quad (k \neq 0)$$ has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. 1. Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha\beta + \beta\gamma + \gamma\alpha\). [2 marks]
   2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 4\). [2 marks]
   3. Explain why \(\alpha^3 - 2\alpha^2 + k = 0\). [1 mark]
   4. Show that \(\alpha^3 + \beta^3 + \gamma^3 = 8 - 3k\). [2 marks]
2. Given that \(\alpha^4 + \beta^4 + \gamma^4 = 0\):
   1. show that \(k = 2\); [4 marks]
   2. find the value of \(\alpha^5 + \beta^5 + \gamma^5\). [3 marks]

1. The arc of the curve \(y^2 = x^2 + 8\) between the points where \(x = 0\) and \(x = 6\) is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = 2\sqrt{2}\pi \int_0^6 \sqrt{x^2 + 4} \, dx$$ [5 marks]
2. By means of the substitution \(x = 2 \sinh \theta\), show that $$S = \pi(24\sqrt{5} + 4\sqrt{2} \sinh^{-1} 3)$$ [8 marks]

1. Show that $$(k + 1)(4(k + 1)^2 - 1) = 4k^3 + 12k^2 + 11k + 3$$ [2 marks]
2. Prove by induction that, for all integers \(n \geqslant 1\), $$1^2 + 3^2 + 5^2 + \ldots + (2n - 1)^2 = \frac{1}{3}n(4n^2 - 1)$$ [6 marks]

1. 1. Use de Moivre's Theorem to show that $$\cos 5\theta = \cos^5 \theta - 10 \cos^3 \theta \sin^2 \theta + 5 \cos \theta \sin^4 \theta$$ and find a similar expression for \(\sin 5\theta\). [5 marks]
   2. Deduce that $$\tan 5\theta = \frac{\tan \theta(5 - 10 \tan^2 \theta + \tan^4 \theta)}{1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$$ [3 marks]
2. Explain why \(t = \tan \frac{\pi}{5}\) is a root of the equation $$t^4 - 10t^2 + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form. [3 marks]
3. Deduce that $$\tan \frac{\pi}{5} \tan \frac{2\pi}{5} = \sqrt{5}$$ [5 marks]