OCR FP1 (Further Pure Mathematics 1) 2008 January

1 The transformation S is a shear with the \(y\)-axis invariant (i.e. a shear parallel to the \(y\)-axis). It is given that the image of the point \(( 1,1 )\) is the point \(( 1,0 )\).

1. Draw a diagram showing the image of the unit square under the transformation S .
2. Write down the matrix that represents S .

2 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 2 } + b \right) \equiv n \left( 2 n ^ { 2 } + 3 n - 2 \right)\), find the values of the constants \(a\) and \(b\).

3 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 24 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence, or otherwise, find the value of \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\).

4 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + 5 z ^ { * }\),
2. \(( z - \mathrm { i } ) ^ { 2 }\),
3. \(\frac { 3 } { z }\).

5 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l } 3
1
2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l } 4
0
3 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { l l l } 2 & 4 & - 1 \end{array} \right)\). Find

1. \(\mathbf { A } - 4 \mathbf { B }\),
2. BC ,
3. CA .

6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = | z - 4 \mathbf { i } | \quad \text { and } \quad \arg z = \frac { 1 } { 6 } \pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence find, in the form \(x +\) i \(y\), the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c } a & 3
- 2 & 1 \end{array} \right)\).

1. Given that \(\mathbf { A }\) is singular, find \(a\).
2. Given instead that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the simultaneous equations $$\begin{aligned} a x + 3 y & = 1
   - 2 x + y & = - 1 \end{aligned}$$

8 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = u _ { n } + 2 n + 1\).

1. Show that \(u _ { 4 } = 16\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

9

1. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )\).
2. The quadratic equation \(x ^ { 2 } - 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\).
3. Show that \(\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
4. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
5. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
6. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }\), find the value of \(N\).