Edexcel F1 (Further Pure Mathematics 1) 2016 January

1. $$z = 3 + 2 \mathrm { i } , \quad w = 1 - \mathrm { i }$$ Find in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z w\)
2. \(\frac { z } { w ^ { * } }\), showing clearly how you obtained your answer. Given that $$| z + k | = \sqrt { 53 } \text {, where } k \text { is a real constant }$$
3. find the possible values of \(k\).

2. $$\mathrm { f } ( x ) = x ^ { 2 } - \frac { 3 } { \sqrt { x } } - \frac { 4 } { 3 x ^ { 2 } } , \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.6,1.7]
2. Taking 1.6 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\). Give your answer to 3 decimal places.

3. The quadratic equation $$x ^ { 2 } - 2 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,

1. 1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
   2. show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 2\)
   3. find the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. 1. show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }\)
   2. find a quadratic equation which has roots $$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.

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

1. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A }\).
2. Hence find the smallest positive integer value of \(n\) for which $$\mathbf { A } ^ { n } = \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. The transformation represented by the matrix \(\mathbf { A }\) followed by the transformation represented by the matrix \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\). Given that \(\mathbf { C } = \left( \begin{array} { r r } 2 & 4
   - 3 & - 5 \end{array} \right)\),
3. find the matrix \(\mathbf { B }\).

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 3 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 3 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be found. Given that $$\sum _ { r = 5 } ^ { 10 } \left( 8 r ^ { 3 } - 3 r + k r ^ { 2 } \right) = 22768$$ (b) find the exact value of the constant \(k\).

6. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant. The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Show that the normal to \(H\) at \(P\) has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
2. Find, in terms of \(c\) and \(p\), the coordinates of \(Q\).

7. $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 15 x ^ { 2 } + 99 x - 130$$

1. Given that \(x = 3 + 2 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of the equation \(\mathrm { f } ( x ) = 0\)
2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

8. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(P\). The point \(B\), which does not lie on the parabola, has coordinates ( \(q , r\) ) where \(q\) and \(r\) are positive constants and \(q > a\). The line \(l\) passes through \(B\) and \(S\).

1. Show that an equation of the line \(l\) is $$( q - a ) y = r ( x - a )$$ The line \(l\) intersects the directrix of \(P\) at the point \(C\). Given that the area of triangle \(O C S\) is three times the area of triangle \(O B S\), where \(O\) is the origin,
2. show that the area of triangle \(O B C\) is \(\frac { 6 } { 5 } \mathrm { qr }\)

9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21