Edexcel FP1 (Further Pure Mathematics 1) 2010 June

1. $$z = 2 - 3 \mathrm { i }$$

1. Show that \(z ^ { 2 } = - 5 - 12 \mathrm { i }\). Find, showing your working,
2. the value of \(\left| z ^ { 2 } \right|\),
3. the value of \(\arg \left( z ^ { 2 } \right)\), giving your answer in radians to 2 decimal places.
4. Show \(z\) and \(z ^ { 2 }\) on a single Argand diagram.

2. \(\mathbf { M } = \left( \begin{array} { c c } 2 a & 3
6 & a \end{array} \right)\), where \(a\) is a real constant.

1. Given that \(a = 2\), find \(\mathbf { M } ^ { - 1 }\).
2. Find the values of \(a\) for which \(\mathbf { M }\) is singular.

3. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 7 } { x } + 2 , \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between 1.4 and 1.5
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2. Starting with the interval [1.4,1.5], use interval bisection twice to find an interval of width 0.025 that contains \(\alpha\).
3. Taking 1.45 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x ) = x ^ { 3 } - \frac { 7 } { x } + 2\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.

4. $$f ( x ) = x ^ { 3 } + x ^ { 2 } + 44 x + 150$$ Given that \(\mathrm { f } ( x ) = ( x + 3 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Find the three roots of \(\mathrm { f } ( x ) = 0\).
3. Find the sum of the three roots of \(\mathrm { f } ( x ) = 0\).

5. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\).

1. Verify that the point \(P \left( 5 t ^ { 2 } , 10 t \right)\) is a general point on \(C\). The point \(A\) on \(C\) has parameter \(t = 4\).
   The line \(l\) passes through \(A\) and also passes through the focus of \(C\).
2. Find the gradient of \(l\).

6. Write down the \(2 \times 2\) matrix that represents

1. an enlargement with centre \(( 0,0 )\) and scale factor 8 ,
2. a reflection in the \(x\)-axis. Hence, or otherwise,
3. find the matrix \(\mathbf { T }\) that represents an enlargement with centre ( 0,0 ) and scale factor 8, followed by a reflection in the \(x\)-axis. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 1
   4 & 2 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } k & 1
   c & - 6 \end{array} \right) , \text { where } k \text { and } c \text { are constants. }$$
4. Find \(\mathbf { A B }\). Given that \(\mathbf { A B }\) represents the same transformation as \(\mathbf { T }\),
5. find the value of \(k\) and the value of \(c\).

7. $$f ( n ) = 2 ^ { n } + 6 ^ { n }$$

1. Show that \(\mathrm { f } ( k + 1 ) = 6 \mathrm { f } ( k ) - 4 \left( 2 ^ { k } \right)\).
2. Hence, or otherwise, prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )\) is divisible by 8 .

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where c is a positive constant. The point \(A\) on \(H\) has \(x\)-coordinate \(3 c\).

1. Write down the \(y\)-coordinate of \(A\).
2. Show that an equation of the normal to \(H\) at \(A\) is $$3 y = 27 x - 80 c$$ The normal to \(H\) at \(A\) meets \(H\) again at the point \(B\).
3. Find, in terms of \(c\), the coordinates of \(B\).

9. (a) Prove by induction that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\),
(b) show that $$\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } + a n + b \right) ,$$ where \(a\) and \(b\) are integers to be found.
(c) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( 7 n ^ { 2 } + 27 n + 26 \right)$$