OCR MEI FP1 (Further Pure Mathematics 1) 2014 June

1 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r ( r - 2 )\), factorising your answer as far as possible.

2 Fig. 2 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h] \end{figure}

1. Write down the matrix \(\mathbf { T }\) representing this transformation. The quadrilateral \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is reflected in the \(x\)-axis to give a new quadrilateral, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
2. Write down the matrix representing reflection in the \(x\)-axis.
3. Find the single matrix that will transform OABC onto \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).

3 You are given that \(z = 2 + 3 \mathrm { j }\) is a root of the quartic equation \(z ^ { 4 } - 5 z ^ { 3 } + 15 z ^ { 2 } - 5 z - 26 = 0\). Find the other roots.

4 Use the identity \(\frac { 1 } { 2 r + 3 } - \frac { 1 } { 2 r + 5 } \equiv \frac { 2 } { ( 2 r + 3 ) ( 2 r + 5 ) }\) and the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 3 ) ( 2 r + 5 ) }\), expressing your answer as a single fraction.

5 The roots of the cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha - 1,3 \beta - 1\) and \(3 \gamma - 1\), expressing your answer in a form with integer coefficients.

6 Prove by induction that \(\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) } = \frac { n } { 2 n + 1 }\).

7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.

1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
3. Sketch the curve.
4. Find the set of values of \(x\) for which \(y > 0\).

8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).

1. Express \(w\) in modulus-argument form.
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).

9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & 3 & - 1
- 1 & \alpha & - 1
- 2 & - 1 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } 3 \alpha - 1 & - 8 & \alpha - 3
5 & 1 & 2
2 \alpha + 1 & - 5 & \alpha + 3 \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } \gamma & 0 & 0
\beta & \gamma & 0
0 & 0 & \gamma \end{array} \right)\).

1. Show that \(\beta = 0\).
2. Find \(\gamma\) in terms of \(\alpha\).
3. Write down \(\mathbf { A } ^ { - 1 }\) for the case when \(\alpha = 2\). State the value of \(\alpha\) for which \(\mathbf { A } ^ { - 1 }\) does not exist.
4. Use your answer to part (iii) to solve the following simultaneous equations. $$\begin{aligned} x + 3 y - z & = 25
   - x + 2 y - z & = 11
   - 2 x - y + 3 z & = - 23 \end{aligned}$$