AQA FP2 (Further Pure Mathematics 2) 2007 June

1

1. Given that \(\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }\), show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
2. Use the method of differences to find the value of $$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$ (4 marks)

2 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
2. Given that \(p\) and \(q\) are real and that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12\) :
   1. explain why the cubic equation has two non-real roots and one real root;
   2. find the value of \(p\).
3. One root of the cubic equation is \(- 1 + 3 \mathrm { i }\). Find:
   1. the other two roots;
   2. the value of \(q\).

3 Use De Moivre's Theorem to find the smallest positive angle \(\theta\) for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$ (5 marks)

4

1. Differentiate \(x \tan ^ { - 1 } x\) with respect to \(x\).
2. Show that $$\int _ { 0 } ^ { 1 } \tan ^ { - 1 } x \mathrm {~d} x = \frac { \pi } { 4 } - \ln \sqrt { 2 }$$ (5 marks)

5 The sketch shows an Argand diagram. The points \(A\) and \(B\) represent the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) respectively. The angle \(A O B = 90 ^ { \circ }\) and \(O A = O B\).
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1. Explain why \(z _ { 2 } = \mathrm { i } z _ { 1 }\).
2. On a single copy of the diagram, draw:
   1. the locus \(L _ { 1 }\) of points satisfying \(\left| z - z _ { 2 } \right| = \left| z - z _ { 1 } \right|\);
   2. the locus \(L _ { 2 }\) of points satisfying \(\arg \left( z - z _ { 2 } \right) = \arg z _ { 1 }\).
3. Find, in terms of \(z _ { 1 }\), the complex number representing the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).

6

1. Show that $$\left( 1 - \frac { 1 } { ( k + 1 ) ^ { 2 } } \right) \times \frac { k + 1 } { 2 k } = \frac { k + 2 } { 2 ( k + 1 ) }$$
2. Prove by induction that for all integers \(n \geqslant 2\) $$\left( 1 - \frac { 1 } { 2 ^ { 2 } } \right) \left( 1 - \frac { 1 } { 3 ^ { 2 } } \right) \left( 1 - \frac { 1 } { 4 ^ { 2 } } \right) \ldots \left( 1 - \frac { 1 } { n ^ { 2 } } \right) = \frac { n + 1 } { 2 n }$$

7 A curve has equation \(y = 4 \sqrt { x }\).

1. Show that the length of arc \(s\) of the curve between the points where \(x = 0\) and \(x = 1\) is given by $$s = \int _ { 0 } ^ { 1 } \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x$$
2. 1. Use the substitution \(x = 4 \sinh ^ { 2 } \theta\) to show that $$\int \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x = \int 8 \cosh ^ { 2 } \theta \mathrm {~d} \theta$$
   2. Hence show that $$s = 4 \sinh ^ { - 1 } 0.5 + \sqrt { 5 }$$

8

1. 1. Given that \(z ^ { 6 } - 4 z ^ { 3 } + 8 = 0\), show that \(z ^ { 3 } = 2 \pm 2 \mathrm { i }\).
   2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. Show that, for any real values of \(k\) and \(\theta\), $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
3. Express \(z ^ { 6 } - 4 z ^ { 3 } + 8\) as the product of three quadratic factors with real coefficients.