OCR FP1 (Further Pure Mathematics 1) 2006 June

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

1. Find \(\mathbf { A } + 3 \mathbf { B }\).
2. Show that \(\mathbf { A } - \mathbf { B } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix and \(k\) is a constant whose value should be stated. \end{itemize}

2 The transformation S is a shear parallel to the \(x\)-axis in which the image of the point ( 1,1 ) is the point \(( 0,1 )\).

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

3 One root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real, is the complex number 2-3i.

1. Write down the other root.
2. Find the values of \(p\) and \(q\).

4 Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + r ^ { 2 } \right) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$

5 The complex numbers \(3 - 2 \mathrm { i }\) and \(2 + \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain these answers,

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

6 In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.

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

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
0 & 1 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 2 }\) and \(\mathbf { A } ^ { 3 }\).
2. Hence suggest a suitable form for the matrix \(\mathbf { A } ^ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2
1 & a & 0
1 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. Hence find the values of \(a\) for which \(\mathbf { M }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + 4 y + 2 z & = 3 a
   x + a y & = 1
   x + 2 y + z & = 3 \end{aligned}$$ have any solutions when
   (a) \(a = 3\),
   (b) \(a = 2\).

9

1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 3 } - r ^ { 3 } \right\} = ( n + 1 ) ^ { 3 } - 1$$
2. Show that \(( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1\).
3. Use the results in parts (i) and (ii) and the standard result for \(\sum _ { r = 1 } ^ { n } r\) to show that $$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$

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

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
2. Find the value of \(p\).
3. Find the value of \(q\).