OCR MEI FP1 (Further Pure Mathematics 1) 2006 June

1

1. State the transformation represented by the matrix \(\left( \begin{array} { r r } 1 & 0
   0 & - 1 \end{array} \right)\).
2. Write down the \(2 \times 2\) matrix for rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
3. Find the \(2 \times 2\) matrix for rotation through \(90 ^ { \circ }\) anticlockwise about the origin, followed by reflection in the \(x\)-axis.

2 Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2 x ^ { 3 } - 3 x ^ { 2 } + x - 2 \equiv ( x + 2 ) \left( A x ^ { 2 } + B x + C \right) + D .$$

3 The cubic equation \(z ^ { 3 } + 4 z ^ { 2 } - 3 z + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 22\).

4 Indicate, on separate Argand diagrams,

1. the set of points \(z\) for which \(| z - ( 3 - \mathrm { j } ) | \leqslant 3\),
2. the set of points \(z\) for which \(1 < | z - ( 3 - \mathrm { j } ) | \leqslant 3\),
3. the set of points \(z\) for which \(\arg ( z - ( 3 - \mathrm { j } ) ) = \frac { 1 } { 4 } \pi\).

5

1. The matrix \(\mathbf { S } = \left( \begin{array} { l l } - 1 & 2
   - 3 & 4 \end{array} \right)\) represents a transformation.
   (A) Show that the point \(( 1,1 )\) is invariant under this transformation.
   (B) Calculate \(\mathbf { S } ^ { - 1 }\).
   (C) Verify that \(( 1,1 )\) is also invariant under the transformation represented by \(\mathbf { S } ^ { - 1 }\).
2. Part (i) may be generalised as follows. If \(( x , y )\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf { T }\), it is also invariant under the transformation represented by \(\mathbf { T } ^ { - 1 }\). Starting with \(\mathbf { T } \binom { x } { y } = \binom { x } { y }\), or otherwise, prove this result.

6 Prove by induction that \(3 + 6 + 12 + \ldots + 3 \times 2 ^ { n - 1 } = 3 \left( 2 ^ { n } - 1 \right)\) for all positive integers \(n\).

9

1. Show that \(r ( r + 1 ) ( r + 2 ) - ( r - 1 ) r ( r + 1 ) \equiv 3 r ( r + 1 )\).
2. Hence use the method of differences to find an expression for \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\).
3. Show that you can obtain the same expression for \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\) using the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\).