OCR MEI FP1 (Further Pure Mathematics 1) 2010 January

1 Two complex numbers are given by \(\alpha = - 3 + \mathrm { j }\) and \(\beta = 5 - 2 \mathrm { j }\).
Find \(\alpha \beta\) and \(\frac { \alpha } { \beta }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.

2 You are given that \(\mathbf { A } = \left( \begin{array} { r } 4
- 2
4 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 5 & 1
2 & - 3 \end{array} \right) , \mathbf { C } = \left( \begin{array} { l l l } 5 & 1 & 8 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { r r } - 2 & 0
4 & 1 \end{array} \right)\).

1. Calculate, where they exist, \(\mathbf { A B } , \mathbf { C A } , \mathbf { B } + \mathbf { D }\) and \(\mathbf { A C }\) and indicate any that do not exist.
2. Matrices \(\mathbf { B }\) and \(\mathbf { D }\) represent transformations B and D respectively. Find the single matrix that represents transformation B followed by transformation D.

3 The roots of the cubic equation \(4 x ^ { 3 } - 12 x ^ { 2 } + k x - 3 = 0\) may be written \(a - d , a\) and \(a + d\). Find the roots and the value of \(k\).

4 You are given that if \(\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1
- 6 & 1 & 1
5 & 2 & 5 \end{array} \right)\) then \(\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1
- 35 & - 15 & 10
17 & 8 & - 4 \end{array} \right)\).
Find the value of \(k\). Hence solve the following simultaneous equations. $$\begin{aligned} 4 x + z & = 9
- 6 x + y + z & = 32
5 x + 2 y + 5 z & = 81 \end{aligned}$$

5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r - 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } - 19 \right)\).

6 Prove by induction that \(1 \times 2 + 2 \times 3 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }\) for all positive integers \(n\). Section B (36 marks)

7 A curve has equation \(y = \frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) }\).

1. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your answers.
3. Sketch the curve.
4. Solve the inequality \(\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0\).

8

1. Fig. 8 shows an Argand diagram. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{df275813-15de-496f-9742-427a9e03f431-3_892_899_1048_664} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure}
   1. Write down the equation of the locus represented by the circumference of circle B.
   2. Write down the two inequalities that define the shaded region between, but not including, circles A and B.
2. 1. Draw an Argand diagram to show the region where $$\frac { \pi } { 4 } < \arg ( z - ( 2 + \mathrm { j } ) ) < \frac { 3 \pi } { 4 }$$
   2. Determine whether the point \(43 + 47 \mathrm { j }\) lies within this region.

9

1. Verify that \(\frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 2 } { r } - \frac { 3 } { r + 1 } + \frac { 1 } { r + 2 }\).
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 3 } { 2 } - \frac { 2 } { n + 1 } + \frac { 1 } { n + 2 } .$$
3. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\) converges as \(n\) tends to infinity.
4. Find \(\sum _ { r = 50 } ^ { 100 } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\), giving your answer to 3 significant figures.