OCR FP1 (Further Pure Mathematics 1) 2014 June

1 Find the determinant of the matrix \(\left( \begin{array} { r r r } a & 4 & - 1
3 & a & 2
a & 1 & 1 \end{array} \right)\).

2 The complex number \(7 + 3 \mathrm { i }\) is denoted by \(z\). Find

1. \(| z |\) and \(\arg z\),
2. \(\frac { z } { 4 - \mathrm { i } }\), showing clearly how you obtain your answer.

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } 2 & 1
- 4 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & 1
2 & 3 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find

1. \(4 \mathbf { A } - \mathbf { B } + 2 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\),
3. \(\left( \mathbf { A B } ^ { - 1 } \right) ^ { - 1 }\).

4

1. Find the matrix that represents a shear with the \(y\)-axis invariant, the image of the point \(( 1,0 )\) being the point \(( 1,4 )\).
2. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 }
   - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\).
   1. Describe fully the geometrical transformation represented by \(\mathbf { X }\).
   2. Find the value of the determinant of \(\mathbf { X }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { X }\).

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

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

6

1. Show that \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 2 ) ^ { 2 } } \equiv \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\).
3. Find \(\sum _ { r = 5 } ^ { \infty } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.

7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(\arg ( z - 2 - 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) and \(| z | = | z - 10 |\) respectively.

1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate, by shading, the region of the Argand diagram for which $$0 \leqslant \arg ( z - 2 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 4 } \pi \text { and } | z | \geqslant | z - 10 | .$$

8

1. Show that \(\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )\).
2. Hence find \(\sum _ { r = n } ^ { 2 n } r \left( r ^ { 2 } - 2 \right)\), giving your answer in a fully factorised form.

9 The roots of the equation \(x ^ { 3 } - k x ^ { 2 } - 2 = 0\) are \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.

1. Show that \(k = \alpha - \frac { 2 } { \alpha ^ { 2 } }\).
2. Given that \(\beta = u + \mathrm { i } v\), where \(u\) and \(v\) are real, find \(u\) in terms of \(\alpha\).
3. Find \(v ^ { 2 }\) in terms of \(\alpha\).

10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 ^ { n } + 2 ^ { n - 1 }\).

1. Find \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Hence suggest a positive integer, other than 1 , which divides exactly into every term of the sequence.
3. By considering \(u _ { n + 1 } + u _ { n }\), prove by induction that your suggestion in part (ii) is correct. \section*{OCR}