Edexcel FP1 (Further Pure Mathematics 1)

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

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 $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 , \text { for } n \geq 1 .$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geq 1\).

7. Given that \(\mathbf { X } = \left( \begin{array} { r r } 2 & a
- 1 & - 1 \end{array} \right)\), 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 \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = \frac { 12 - 5 \mathrm { i } } { z _ { 1 } }\),

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.