Edexcel FP1 (Further Pure Mathematics 1)

1. $$\mathbf { R } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { S } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$

1. Find \(\mathbf { R } ^ { 2 }\).
2. Find \(\mathbf { R S }\).
3. Describe the geometrical transformation represented by \(\mathbf { R S }\).

2. A point \(P\) with coordinates \(( x , y )\) moves so that its distance from the point \(( - 3,0 )\) is equal to its distance from the line \(x = 3\). Find a cartesian equation for the locus of \(P\).

3. \(z = 1 + \mathrm { i } \sqrt { 3 }\) Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.

1. \(z ^ { 2 } + z\),
2. \(\frac { z } { 3 - z }\),
   giving the exact values of \(a\) and \(b\) in each part.

4. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 5 x - 3\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval ( 2,3 ).

1. Use linear interpolation on the end points of this interval to obtain an approximation for \(\alpha\).
2. Taking 2.5 as a first approximation to \(\alpha\), apply the Newton - Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 2 decimal places.

5. Given that \(a\) and \(b\) are non-zero constants and that $$\mathbf { X } = \left( \begin{array} { r r } a & 2 b \\ - a & 3 b \end{array} \right) ,$$

1. find \(\mathbf { X } ^ { - 1 }\), giving your answer in terms of \(a\) and \(b\). Given also that \(\mathbf { Z X } = \mathbf { Y }\), where \(\mathbf { Y } = \left( \begin{array} { c c } 3 a & b \\ a & 2 b \end{array} \right)\),
2. find \(\mathbf { Z }\), simplifying your answer.

6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 6 \right) = \frac { 1 } { 2 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$ (b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( 2 r ^ { 2 } - 6 \right)\).

7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).

1. Find each of these roots in the form \(a + b \mathrm { i }\).
2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
3. Illustrate the two roots on a single Argand diagram.
4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\). The point ( \(3 t , \frac { 3 } { t }\) ) is a general point on this hyperbola.

1. Find the value of \(c ^ { 2 }\).
2. Show that an equation of the normal to \(H\) at the point ( \(3 t , \frac { 3 } { t }\) ) is $$y = t ^ { 2 } x + \left( \frac { 3 } { t } - 3 t ^ { 3 } \right)$$ The point \(P\) on \(H\) has coordinates (6, 1.5). The tangent to \(H\) at \(P\) meets the curve again at the point \(Q\).
3. Find the coordinates of the point \(Q\).

9. (a) A sequence of numbers is defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = 3 u _ { n } + 4 \text { for } n \geqslant 1 .$$ Prove by induction that $$u _ { n } = 3 ^ { n } + 2 \left( 3 ^ { n - 1 } - 1 \right) \text { for } n \in \mathbb { Z } ^ { + } \text {. }$$ (b) $$\mathbf { A } = \left( \begin{array} { l l } 4 & 0 \\ 9 & 1 \end{array} \right)$$

1. Prove by induction that $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right) \text { for } n \in \mathbb { Z } ^ { + } .$$
2. Determine whether the result \(\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right)\) is also valid for \(n = - 1\).