OCR FP1 (Further Pure Mathematics 1) 2009 June

1 Evaluate \(\sum _ { r = 101 } ^ { 250 } r ^ { 3 }\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 5 & 0
0 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find the values of the constants \(a\) and \(b\) for which \(a \mathbf { A } + b \mathbf { B } = \mathbf { I }\).

3 The complex numbers \(z\) and \(w\) are given by \(z = 5 - 2 \mathrm { i }\) and \(w = 3 + 7 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find

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

4 The roots of the quadratic equation \(x ^ { 2 } + x - 8 = 0\) are \(p\) and \(q\). Find the value of \(p + q + \frac { 1 } { p } + \frac { 1 } { q }\).

5 The cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Use the substitution \(x = \sqrt { u }\) to find a cubic equation in \(u\) with integer coefficients.
2. Hence find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).

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

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   (a) \(| z - a | = 3 \sqrt { 2 }\),
   (b) \(\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi\).
3. Indicate, by shading, the region of the Argand diagram for which $$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$

7

1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 4 } - r ^ { 4 } \right\} = ( n + 1 ) ^ { 4 } - 1$$
2. Show that \(( r + 1 ) ^ { 4 } - r ^ { 4 } \equiv 4 r ^ { 3 } + 6 r ^ { 2 } + 4 r + 1\).
3. Hence show that $$4 \sum _ { r = 1 } ^ { n } r ^ { 3 } = n ^ { 2 } ( n + 1 ) ^ { 2 }$$

8 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 3 & 2
1 & 1 \end{array} \right)\).

1. Draw a diagram showing the image of the unit square under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a transformation S followed by another transformation T.
2. Given that S is a shear with the \(y\)-axis invariant in which the image of the point ( 1,1 ) is ( 1,2 ), write down the matrix that represents \(S\).
3. Find the matrix that represents transformation T and describe fully the transformation T .

9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 1 & 1
1 & a & 1
1 & 1 & 2 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + y + z & = 2 a
   x + a y + z & = - 1
   x + y + 2 z & = - 1 \end{aligned}$$ have any solutions when
   (a) \(a = 0\),
   (b) \(a = 1\).

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

1. Find \(u _ { 2 }\) and \(u _ { 3 }\) and verify that \(\frac { 1 } { 2 } \left( u _ { 4 } - 1 \right) = 27\).
2. Hence suggest an expression for \(u _ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.