Edexcel F1 (Further Pure Mathematics 1) 2015 January

1. $$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$ Given that \(x = 1 + 2 \mathrm { i }\) 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. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + \frac { 1 } { 2 \sqrt { x ^ { 5 } } } + 2 , \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 2,3 ]\).
2. Taking 3 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. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

4. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 12 x\) The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on \(C\), where \(p \neq 0\)

1. Show that the equation of the normal to the curve \(C\) at the point \(P\) is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve \(C\) again at the point \(Q\).
   Given that \(p = 2\) and that \(S\) is the focus of the parabola, find
2. the coordinates of the point \(Q\),
3. the area of the triangle \(P Q S\).

5. The quadratic equation $$4 x ^ { 2 } + 3 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

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

6.

1. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 0
   0 & 1 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 }
   - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$ (a) Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
   (b) Describe fully the single transformation represented by the matrix \(\mathbf { B }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
   (c) Find \(\mathbf { C }\).
2. \(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4
   1 & k \end{array} \right)\), where \(k\) is a real number. Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).

7. Given that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } ( r + a ) ( r + b ) = \frac { 1 } { 6 } n ( 2 n + 11 ) ( n - 1 )$$ where \(a\) and \(b\) are constants and \(a > b\),

1. find the value of \(a\) and the value of \(b\).
2. Find the value of $$\sum _ { r = 9 } ^ { 20 } ( r + a ) ( r + b )$$

1. (i) A sequence of numbers is defined by

$$\begin{gathered} u _ { 1 } = 5 \quad u _ { 2 } = 13
u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 ^ { n } + 3 ^ { n }$$ (ii) Prove by induction that for \(n \geqslant 2\), where \(n \in \mathbb { Z }\), $$f ( n ) = 7 ^ { 2 n } - 48 n - 1$$ is divisible by 2304
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