OCR FP1 (Further Pure Mathematics 1) 2011 January

\(\mathbf { 1 }\) The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & - 1 \end{array} \right)\) and \(\mathbf { C } = \binom { 4 } { 2 }\). Find

1. \(2 \mathbf { A } + \mathbf { B }\),
2. \(\mathbf { A C }\),
3. CB. \end{itemize}

2 The complex numbers \(z\) and \(w\) are given by \(z = 4 + 3 \mathrm { i }\) and \(w = 6 - \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find

1. \(3 z - 4 w\),
2. \(\frac { z ^ { * } } { w }\).

3 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 2\), and \(u _ { n + 1 } = 2 u _ { n } - 1\) for \(n \geqslant 1\). Prove by induction that \(u _ { n } = 2 ^ { n - 1 } + 1\).

4 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 3 } + b r \right) \equiv n ( n - 1 ) ( n + 1 ) ( n + 2 )\), find the values of the constants \(a\) and \(b\).

5 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are non-singular square matrices, simplify $$\mathbf { A B } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) ^ { - 1 } .$$

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1. Sketch on a single Argand diagram the loci given by
   (a) \(\quad | z | = | z - 8 |\),
   (b) \(\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
2. Indicate by shading the region of the Argand diagram for which $$| z | \leqslant | z - 8 | \quad \text { and } \quad 0 \leqslant \arg ( z + 2 i ) \leqslant \frac { 1 } { 4 } \pi$$

1. Write down the matrix, \(\mathbf { A }\), that represents a shear with \(x\)-axis invariant in which the image of the point \(( 1,1 )\) is \(( 4,1 )\).
2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0
   0 & \sqrt { 3 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
3. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 2 & 6
   0 & 2 \end{array} \right)\).
   (a) Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
   (b) Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\). 8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).
4. Show that \(p = \frac { 5 } { 6 }\).
5. Find the value of \(q\). 9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1
   3 & a & 1
   4 & 2 & 1 \end{array} \right)\).
6. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
7. Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
8. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k
   & 3 x + 6 y + z = 0
   & 4 x + 2 y + z = k \end{aligned}$$ where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
9. Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
10. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
11. Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).

8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).

1. Show that \(p = \frac { 5 } { 6 }\).
2. Find the value of \(q\).

9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1
3 & a & 1
4 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
3. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k
   & 3 x + 6 y + z = 0
   & 4 x + 2 y + z = k \end{aligned}$$ where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
4. Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
5. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
6. Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).

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   ACHIEVEMENT