OCR FP1 (Further Pure Mathematics 1) 2013 January

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 1
1 & 4 \end{array} \right)\), where \(a \neq \frac { 1 } { 4 }\), and \(\mathbf { I }\) denotes the \(2 \times 2\) identity matrix. Find

1. \(2 \mathbf { A } - 3 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\).

2 Find \(\sum _ { r = 1 } ^ { n } ( r - 1 ) ( r + 1 )\), giving your answer in a fully factorised form.

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

1. Find \(| z |\) and \(\arg z\).
2. Given that \(a z + b z ^ { * } = 4 - 8 \mathrm { i }\), find the values of the real constants \(a\) and \(b\).

4 The quadratic equation \(x ^ { 2 } + x + k = 0\) has roots \(\alpha\) and \(\beta\).

1. Use the substitution \(x = 2 u + 1\) to obtain a quadratic equation in \(u\).
2. Hence, or otherwise, find the value of \(\left( \frac { \alpha - 1 } { 2 } \right) \left( \frac { \beta - 1 } { 2 } \right)\) in terms of \(k\).

5 By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations $$\begin{array} { r } 3 x + 2 y + 4 z = 5
\lambda y + z = 1
x + \lambda y + \lambda z = 4 \end{array}$$ do not have a unique solution for \(x , y\) and \(z\).
\includegraphics[max width=\textwidth, alt={}, center]{f074de40-08b6-47a6-a0d2-d3cbe628cacc-3_556_759_233_653} The diagram shows the unit square \(O A B C\), and its image \(O A B ^ { \prime } C ^ { \prime }\) after a transformation. The points have the following coordinates: \(A ( 1,0 ) , B ( 1,1 ) , C ( 0,1 ) , B ^ { \prime } ( 3,2 )\) and \(C ^ { \prime } ( 2,2 )\).

1. Write down the matrix, \(\mathbf { X }\), for this transformation.
2. The transformation represented by \(\mathbf { X }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them.
3. Find the matrix that represents transformation Q followed by transformation P .

7

1. Sketch on a single Argand diagram the loci given by
   (a) \(| z | = 2\),
   (b) \(\quad \arg ( z - 3 - \mathrm { i } ) = \pi\).
2. Indicate, by shading, the region of the Argand diagram for which $$| z | \leqslant 2 \text { and } 0 \leqslant \arg ( z - 3 - i ) \leqslant \pi .$$

8

1. Show that \(\frac { 1 } { r } - \frac { 3 } { r + 1 } + \frac { 2 } { r + 2 } \equiv \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) }\).
2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) } = \frac { n } { ( n + 1 ) ( n + 2 ) }\).
3. Find the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) }\).

9

1. Show that \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 } \equiv \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + 2 \alpha \beta \gamma ( \alpha + \beta + \gamma )\).
2. It is given that \(\alpha , \beta\) and \(\gamma\) are the roots of the cubic equation \(x ^ { 3 } + p x ^ { 2 } - 4 x + 3 = 0\), where \(p\) is a constant. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) in terms of \(p\).

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

1. Find \(u _ { 2 }\) and \(u _ { 3 }\), and show that \(u _ { 4 } = \frac { 2 } { 7 }\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.