OCR FP1 (Further Pure Mathematics 1) 2007 January

\(\mathbf { 1 }\) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1
3 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } a & - 1
- 3 & - 2 \end{array} \right)\).

1. Given that \(2 \mathbf { A } + \mathbf { B } = \left( \begin{array} { l l } 1 & 1
   3 & 2 \end{array} \right)\), write down the value of \(a\).
2. Given instead that \(\mathbf { A B } = \left( \begin{array} { l l } 7 & - 4
   9 & - 7 \end{array} \right)\), find the value of \(a\). \end{itemize}

2 Use an algebraic method to find the square roots of the complex number \(15 + 8 \mathrm { i }\).

3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to find $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$ expressing your answer in a fully factorised form.

4

1. Sketch, on an Argand diagram, the locus given by \(| z - 1 + \mathrm { i } | = \sqrt { 2 }\).
2. Shade on your diagram the region given by \(1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }\).

5

1. Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
2. Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
3. Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).

6 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = n ^ { 2 } + 3 n\), for all positive integers \(n\).

1. Show that \(u _ { n + 1 } - u _ { n } = 2 n + 4\).
2. Hence prove by induction that each term of the sequence is divisible by 2 .

7 The quadratic equation \(x ^ { 2 } + 5 x + 10 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 5\).
3. Hence find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).

8

1. Show that \(( r + 2 ) ! - ( r + 1 ) ! = ( r + 1 ) ^ { 2 } \times r !\).
2. Hence find an expression, in terms of \(n\), for $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots + ( n + 1 ) ^ { 2 } \times n ! .$$
3. State, giving a brief reason, whether the series $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots$$ converges.

9 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 0 & 3
- 1 & 0 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a rotation, R , followed by another transformation, S.
2. Describe fully the rotation R and write down the matrix that represents R .
3. Describe fully the transformation S and write down the matrix that represents S .

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

1. Find \(\mathbf { D } ^ { - 1 }\).
2. Hence, or otherwise, solve the equations $$\begin{aligned} a x + 2 y & = 3
   3 x + y + 2 z & = 4
   - y + z & = 1 \end{aligned}$$