OCR FP1 (Further Pure Mathematics 1) 2011 June

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 2 & a \\ 4 & 1 \end{array} \right)\). I denotes the \(2 \times 2\) identity matrix. Find

1. \(\mathbf { A } + 3 \mathbf { B } - 4 \mathbf { I }\),
2. AB.

2 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).

3 By using the determinant of an appropriate matrix, find the values of \(k\) for which the simultaneous equations $$\begin{aligned} & k x + 8 y = 1 \\ & 2 x + k y = 3 \end{aligned}$$ do not have a unique solution.

4 Find \(\sum _ { r = 1 } ^ { 2 n } \left( 3 r ^ { 2 } - \frac { 1 } { 2 } \right)\), expressing your answer in a fully factorised form.

5 The complex number \(1 + \mathrm { i } \sqrt { 3 }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by \(| z - a | = | a |\) and \(\arg ( z - a ) = \frac { 1 } { 2 } \pi\).

6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r r } a & 1 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)\), where \(a \neq 1\). Find \(\mathbf { C } ^ { - 1 }\).

7

1. Show that \(\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { 2 } { r ^ { 2 } - 1 }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 2 } { r ^ { 2 } - 1 }\).
3. Find the value of \(\sum _ { r = 1000 } ^ { \infty } \frac { 2 } { r ^ { 2 } - 1 }\).

8 The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 0 & 3 \\ 3 & 0 \end{array} \right)\).

1. The diagram in the printed answer book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { X }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. The transformation represented by \(\mathbf { X }\) is equivalent to a transformation A , followed by a transformation B. Give geometrical descriptions of possible transformations A and B and state the matrices that represent them.

9 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is \(16 - 30 \mathrm { i }\).

1. Write down the other root of the quadratic equation.
2. Find the values of \(a\) and \(b\).
3. Use an algebraic method to solve the quartic equation \(y ^ { 4 } + a y ^ { 2 } + b = 0\).

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

1. Use the substitution \(x = \frac { 1 } { \sqrt { u } }\) to show that \(4 u ^ { 3 } + 12 u ^ { 2 } + 9 u - 1 = 0\).
2. Hence find the values of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) and \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).