Edexcel FP1 (Further Pure Mathematics 1) 2010 January

1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by

$$z _ { 1 } = 2 + 8 i \quad \text { and } \quad z _ { 2 } = 1 - i$$ Find, showing your working,

1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
2. the value of \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right|\),
3. the value of \(\arg \frac { Z _ { 1 } } { Z _ { 2 } }\), giving your answer in radians to 2 decimal places.

2. $$f ( x ) = 3 x ^ { 2 } - \frac { 11 } { x ^ { 2 } }$$

1. Write down, to 3 decimal places, the value of \(\mathrm { f } ( 1.3 )\) and the value of \(\mathrm { f } ( 1.4 )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between 1.3 and 1.4
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2. Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains \(\alpha\).
3. Taking 1.4 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.

3. A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 2 \\ u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 . \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the parabola with equation \(y ^ { 2 } = 12 x\).
The point \(P\) on the parabola has \(x\)-coordinate \(\frac { 1 } { 3 }\).
The point \(S\) is the focus of the parabola.

1. Write down the coordinates of \(S\). The points \(A\) and \(B\) lie on the directrix of the parabola.
   The point \(A\) is on the \(x\)-axis and the \(y\)-coordinate of \(B\) is positive. Given that \(A B P S\) is a trapezium,
2. calculate the perimeter of \(A B P S\).

5. \(\mathbf { A } = \left( \begin{array} { c c } a & - 5 \\ 2 & a + 4 \end{array} \right)\), where \(a\) is real.

1. Find \(\operatorname { det } \mathbf { A }\) in terms of \(a\).
2. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(a\). Given that \(a = 0\),
3. find \(\mathbf { A } ^ { - 1 }\).

6. Given that 2 and \(5 + 2 \mathrm { i }\) are roots of the equation $$x ^ { 3 } - 12 x ^ { 2 } + c x + d = 0 , \quad c , d \in \mathbb { R }$$

1. write down the other complex root of the equation.
2. Find the value of \(c\) and the value of \(d\).
3. Show the three roots of this equation on a single Argand diagram.

7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).

1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).

8. (a) Prove by induction that, for any positive integer \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + 3 r + 2 \right) = \frac { 1 } { 4 } n ( n + 2 ) \left( n ^ { 2 } + 7 \right)$$ (c) Hence evaluate \(\sum _ { r = 15 } ^ { 25 } \left( r ^ { 3 } + 3 r + 2 \right)\)

9. $$\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$

1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\).
3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
5. Find the coordinates of \(C\).