Edexcel FP1 (Further Pure Mathematics 1) 2012 January

1. Given that \(z _ { 1 } = 1 - \mathrm { i }\),
   1. find \(\arg \left( z _ { 1 } \right)\).
   Given also that \(z _ { 2 } = 3 + 4 \mathrm { i }\), find, in the form \(a + \mathrm { i } b , a , b \in \mathbb { R }\),
2. \(z _ { 1 } z _ { 2 }\),
3. \(\frac { z _ { 2 } } { z _ { 1 } }\). In part (b) and part (c) you must show all your working clearly.

2. (a) Show that \(\mathrm { f } ( x ) = x ^ { 4 } + x - 1\) has a real root \(\alpha\) in the interval [0.5, 1.0].
[0pt] (b) Starting with the interval [0.5, 1.0], use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\).
(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\). Give your answer to 3 decimal places.

3. A parabola \(C\) has cartesian equation \(y ^ { 2 } = 16 x\). The point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).

1. Write down the coordinates of the focus \(F\) and the equation of the directrix of \(C\).
2. Show that the equation of the normal to \(C\) at \(P\) is \(y + t x = 8 t + 4 t ^ { 3 }\).

4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).

1. Find the coordinates of the vertices of \(T ^ { \prime }\).
2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2
   3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2
   3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
3. Find \(\mathbf { Q R }\).
4. Find the determinant of \(\mathbf { Q R }\).
5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).

5. The roots of the equation $$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

1. Given that \(z _ { 1 } = 3 + \mathrm { i }\), find \(z _ { 2 }\) and \(z _ { 3 }\).
2. Show, on a single Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

6. (a) Prove by induction $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the result in part (a), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 2 \right) = \frac { 1 } { 4 } n \left( n ^ { 3 } + 2 n ^ { 2 } + n - 8 \right)$$ (c) Calculate the exact value of \(\sum _ { r = 20 } ^ { 50 } \left( r ^ { 3 } - 2 \right)\).

7. A sequence can be described by the recurrence formula $$u _ { n + 1 } = 2 u _ { n } + 1 , \quad n \geqslant 1 , \quad u _ { 1 } = 1$$

1. Find \(u _ { 2 }\) and \(u _ { 3 }\).
2. Prove by induction that \(u _ { n } = 2 ^ { n } - 1\)

8. $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1
2 & 3 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular.
2. Find \(\mathbf { B }\) such that \(\mathbf { B A } ^ { 2 } = \mathbf { A }\).

9. The rectangular hyperbola \(H\) has cartesian equation \(x y = 9\) The points \(P \left( 3 p , \frac { 3 } { p } \right)\) and \(Q \left( 3 q , \frac { 3 } { q } \right)\) lie on \(H\), where \(p \neq \pm q\).

1. Show that the equation of the tangent at \(P\) is \(x + p ^ { 2 } y = 6 p\).
2. Write down the equation of the tangent at \(Q\). The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
3. Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\).