Edexcel FP1 (Further Pure Mathematics 1) 2009 January

1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.

2. (a) Show, using the formulae for \(\sum r\) and \(\sum r ^ { 2 }\), that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 4 r - 1 \right) = n ( n + 2 ) ( 2 n + 1 )$$ (b) Hence, or otherwise, find the value of \(\sum _ { r = 11 } ^ { 20 } \left( 6 r ^ { 2 } + 4 r - 1 \right)\).

3. The rectangular hyperbola, \(H\), has parametric equations \(x = 5 t , y = \frac { 5 } { t } , t \neq 0\).

1. Write the cartesian equation of \(H\) in the form \(x y = c ^ { 2 }\). Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively.
2. Find the coordinates of the mid-point of \(A B\).

4. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }$$

5. $$f ( x ) = 3 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 20$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.1,1.2].
2. Find \(\mathrm { f } ^ { \prime } ( x )\).
3. Using \(x _ { 0 } = 1.1\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 significant figures.

6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 \text {, for } n \geqslant 1 \text {. }$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geqslant 1\).

7. Given that \(\mathbf { X } = \left( \begin{array} { c c } 2 & a
- 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),

1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 } .$$ This tangent meets the \(y\)-axis at the point \(R\).
2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
3. Show that \(l\) passes through the focus of the parabola.
4. Find the coordinates of the point where \(l\) meets the directrix of the parabola.

9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),

1. find \(z _ { 2 }\) in the form \(a + i b\), where \(a\) and \(b\) are real.
2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
4. the complex number represented by C,
5. the exact value of the radius of the circle.

10. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0
0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1
1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$

1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
2. Find \(\mathbf { D }\).
3. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3
   3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
4. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
5. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).