OCR FP1 (Further Pure Mathematics 1) 2005 June

1 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 2 r + 1 \right) = n \left( 2 n ^ { 2 } + 4 n + 3 \right)$$

2 The matrices \(\mathbf { A }\) and \(\mathbf { I }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2
1 & 3 \end{array} \right)\) and \(\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right)\) respectively.

1. Find \(\mathbf { A } ^ { 2 }\) and verify that \(\mathbf { A } ^ { 2 } = 4 \mathbf { A } - \mathbf { I }\).
2. Hence, or otherwise, show that \(\mathbf { A } ^ { - 1 } = 4 \mathbf { I } - \mathbf { A }\).

3 The complex numbers \(2 + 3 \mathrm { i }\) and \(4 - \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answers.

1. \(z + 5 w\),
2. \(z ^ { * } w\), where \(z ^ { * }\) is the complex conjugate of \(z\),
3. \(\frac { 1 } { w }\).

4 Use an algebraic method to find the square roots of the complex number 21-20i.

5

1. Show that $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 } + \ldots + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
3. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).

6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z - 2 \mathrm { i } | = 2 \quad \text { and } \quad | z + 1 | = | z + \mathrm { i } |$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence write down the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
   \(7 \quad\) The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r r } a & 1 & 3
   2 & 1 & - 1
   0 & 1 & 2 \end{array} \right)\).
3. Given that \(\mathbf { B }\) is singular, show that \(a = - \frac { 2 } { 3 }\).
4. Given instead that \(\mathbf { B }\) is non-singular, find the inverse matrix \(\mathbf { B } ^ { - 1 }\).
5. Hence, or otherwise, solve the equations $$\begin{aligned} - x + y + 3 z & = 1
   2 x + y - z & = 4
   y + 2 z & = - 1 \end{aligned}$$

8

1. The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
   1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
   2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 4\).
   3. Hence find a quadratic equation which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
2. The cubic equation \(x ^ { 3 } - 12 x ^ { 2 } + a x - 48 = 0\) has roots \(p , 2 p\) and \(3 p\).
   1. Find the value of \(p\).
   2. Hence find the value of \(a\).

9

1. Write down the matrix \(\mathbf { C }\) which represents a stretch, scale factor 2 , in the \(x\)-direction.
2. The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l } 1 & 3
   0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { D }\).
3. The matrix \(\mathbf { M }\) represents the combined effect of the transformation represented by \(\mathbf { C }\) followed by the transformation represented by \(\mathbf { D }\). Show that $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3
   0 & 1 \end{array} \right)$$
4. Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right)
   0 & 1 \end{array} \right)\), for all positive integers \(n\).