Edexcel FP1 (Further Pure Mathematics 1) 2014 June

1. The roots of the equation

$$2 z ^ { 3 } - 3 z ^ { 2 } + 8 z + 5 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) Given that \(z _ { 1 } = 1 + 2 i\), find \(z _ { 2 }\) and \(z _ { 3 }\)

2. $$\mathrm { f } ( x ) = 3 \cos 2 x + x - 2 , \quad - \pi \leqslant x < \pi$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [2,3].
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2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
3. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval \([ - 1,0 ]\). Starting with this interval, use interval bisection to find an interval of width 0.25 which contains \(\beta\).

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

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents an enlargement, scale factor - 2 , with centre the origin.
2. Write down the matrix \(\mathbf { B }\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } 3 & k \\ - 2 & 3 \end{array} \right) , \quad \text { where } k \text { is a positive constant. }$$ Triangle \(T\) has an area of 16 square units. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of the triangle \(T ^ { \prime }\) is 224 square units, find the value of \(k\).

4. The complex number \(z\) is given by $$z = \frac { p + 2 \mathrm { i } } { 3 + p \mathrm { i } }$$ where \(p\) is an integer.

1. Express \(z\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\).
2. Given that \(\arg ( z ) = \theta\), where \(\tan \theta = 1\) find the possible values of \(p\).

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that

$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 )$$ (b) Calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( r ^ { 2 } - 3 \right)\)

6. $$\mathbf { A } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } - 1 & 1 \\ 0 & 1 \end{array} \right)$$ Given that \(\mathbf { M } = ( \mathbf { A } + \mathbf { B } ) ( 2 \mathbf { A } - \mathbf { B } )\),

1. calculate the matrix \(\mathbf { M }\),
2. find the matrix \(\mathbf { C }\) such that \(\mathbf { M C } = \mathbf { A }\).

7. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 4 a x , a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(P ^ { \prime } \left( a p ^ { 2 } , - 2 a p \right)\) lie on \(C\).

1. Show that an equation of the normal to \(C\) at the point \(P\) is $$y + p x = 2 a p + a p ^ { 3 }$$
2. Write down an equation of the normal to \(C\) at the point \(P ^ { \prime }\). The normal to \(C\) at \(P\) meets the normal to \(C\) at \(P ^ { \prime }\) at the point \(Q\).
3. Find, in terms of \(a\) and \(p\), the coordinates of \(Q\). Given that \(S\) is the focus of the parabola,
4. find the area of the quadrilateral \(S P Q P ^ { \prime }\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\). An equation for the tangent to \(H\) at \(P\) is given by $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$ The points \(A\) and \(B\) lie on \(H\).
The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)\).
Find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).

9. (a) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n }$$ (b) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 , \quad u _ { 2 } = 32 , \\ u _ { n + 2 } = 6 u _ { n + 1 } - 8 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 4 ^ { n + 1 } - 2 ^ { n + 3 }$$