Edexcel FP1 (Further Pure Mathematics 1) 2009 June

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

$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$

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

2. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + k ) ,$$ where \(k\) is a constant to be found.
(b) Hence evaluate \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\).

3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\).
2. Find the sum of these four roots.

4. Given that \(\alpha\) is the only real root of the equation $$x ^ { 3 } - x ^ { 2 } - 6 = 0$$

1. show that \(2.2 < \alpha < 2.3\)
2. Taking 2.2 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 6\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.
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3. Use linear interpolation once on the interval [2.2, 2.3] to find another approximation to \(\alpha\), giving your answer to 3 decimal places.

5. \(\mathbf { R } = \left( \begin{array} { l l } a & 2
a & b \end{array} \right)\), where \(a\) and \(b\) are constants and \(a > 0\).

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). Given that \(\mathbf { R } ^ { 2 }\) represents an enlargement with centre ( 0,0 ) and scale factor 15 ,
2. find the value of \(a\) and the value of \(b\).

6. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\).

1. Verify that the point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
2. Write down the coordinates of the focus \(S\) of \(C\).
3. Show that the normal to \(C\) at \(P\) has equation $$y + t x = 8 t + 4 t ^ { 3 }$$ The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(N\).
4. Find the area of triangle \(P S N\) in terms of \(t\), giving your answer in its simplest form.

7. \(\mathbf { A } = \left( \begin{array} { r r } a & - 2
- 1 & 4 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { A }\) is singular. $$\mathbf { B } = \left( \begin{array} { r r } 3 & - 2
   - 1 & 4 \end{array} \right)$$
2. Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { B }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 6,3 k + 12 )\), where \(k\) is a constant,
3. show that \(P\) lies on the line with equation \(y = x + 3\).

8. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),

1. \(\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3\) is divisible by 4 ,
2. \(\left( \begin{array} { l l } 3 & - 2
   2 & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { l r } 2 n + 1 & - 2 n
   2 n & 1 - 2 n \end{array} \right)\)