Edexcel FP1 (Further Pure Mathematics 1) 2011 June

1. $$f ( x ) = 3 ^ { x } + 3 x - 7$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 1\) and \(x = 2\).
2. Starting with the interval \([ 1,2 ]\), use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).

2. $$z _ { 1 } = - 2 + \mathrm { i }$$

1. Find the modulus of \(z _ { 1 }\).
2. Find, in radians, the argument of \(z _ { 1 }\), giving your answer to 2 decimal places. The solutions to the quadratic equation $$z ^ { 2 } - 10 z + 28 = 0$$ are \(z _ { 2 }\) and \(z _ { 3 }\).
3. Find \(z _ { 2 }\) and \(z _ { 3 }\), giving your answers in the form \(p \pm i \sqrt { } q\), where \(p\) and \(q\) are integers.
4. Show, on an Argand diagram, the points representing your complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

3. (a) Given that $$\mathbf { A } = \left( \begin{array} { c c } 1 & \sqrt { } 2
\sqrt { } 2 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { 2 }\),
2. describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
   (b) Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 1
   - 1 & 0 \end{array} \right)$$ describe fully the geometrical transformation represented by \(\mathbf { B }\).
   (c) Given that $$\mathbf { C } = \left( \begin{array} { c c } k + 1 & 12
   k & 9 \end{array} \right)$$ where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.

4. $$f ( x ) = x ^ { 2 } + \frac { 5 } { 2 x } - 3 x - 1 , \quad x \neq 0$$

1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.7, 0.9].
2. Taking 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.

5. \(\mathbf { A } = \left( \begin{array} { r r } - 4 & a
b & - 2 \end{array} \right)\), where \(a\) and \(b\) are constants. Given that the matrix \(\mathbf { A }\) maps the point with coordinates \(( 4,6 )\) onto the point with coordinates \(( 2 , - 8 )\),

1. find the value of \(a\) and the value of \(b\). A quadrilateral \(R\) has area 30 square units.
   It is transformed into another quadrilateral \(S\) by the matrix \(\mathbf { A }\).
   Using your values of \(a\) and \(b\),
2. find the area of quadrilateral \(S\).

6. Given that \(z = x + \mathrm { i } y\), find the value of \(x\) and the value of \(y\) such that $$z + 3 \mathrm { i } z ^ { * } = - 1 + 13 \mathrm { i }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

7. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n ( 2 n + 1 ) ( 2 n - 1 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 3 n } ( 2 r - 1 ) ^ { 2 } = \frac { 2 } { 3 } n \left( a n ^ { 2 } + b \right)$$ where \(a\) and \(b\) are integers to be found.

8. The parabola \(C\) has equation \(y ^ { 2 } = 48 x\). The point \(P \left( 12 t ^ { 2 } , 24 t \right)\) is a general point on \(C\).

1. Find the equation of the directrix of \(C\).
2. Show that the equation of the tangent to \(C\) at \(P \left( 12 t ^ { 2 } , 24 t \right)\) is $$x - t y + 12 t ^ { 2 } = 0$$ The tangent to \(C\) at the point \(( 3,12 )\) meets the directrix of \(C\) at the point \(X\).
3. Find the coordinates of \(X\).

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

1. \(\left( \begin{array} { l l } 3 & 0
   6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0
   3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)\),
2. \(\mathrm { f } ( n ) = 7 ^ { 2 n - 1 } + 5\) is divisible by 12 .