OCR MEI FP1 (Further Pure Mathematics 1) 2013 January

1 Transformation A is represented by matrix \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)\) and transformation B is represented by matrix \(\mathbf { B } = \left( \begin{array} { l l } 2 & 0
0 & 3 \end{array} \right)\).

1. Describe transformations A and B .
2. Find the matrix for the composite transformation A followed by B .

2 Given that \(z = a + b \mathrm { j }\), find \(\operatorname { Re } \left( \frac { z } { z ^ { * } } \right)\) and \(\operatorname { Im } \left( \frac { z } { z ^ { * } } \right)\).

3 You are given that \(z = 2 + \mathrm { j }\) is a root of the cubic equation \(2 z ^ { 3 } + p z ^ { 2 } + 22 z - 15 = 0\), where \(p\) is real. Find the other roots and the value of \(p\).

4

1. Show that \(x ^ { 2 } - x + 2 > 0\) for all real \(x\).
2. Solve the inequality \(\frac { 2 x } { x ^ { 2 } - x + 2 } > x\).

5 You are given that \(\frac { 3 } { ( 5 + 3 x ) ( 2 + 3 x ) } \equiv \frac { 1 } { 2 + 3 x } - \frac { 1 } { 5 + 3 x }\).

1. Use this result to find \(\sum _ { r = 1 } ^ { 100 } \frac { 1 } { ( 5 + 3 r ) ( 2 + 3 r ) }\), giving your answer as an exact fraction.
2. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 + 3 r ) ( 2 + 3 r ) }\) converges as \(n\) tends to infinity.

6 Prove by induction that \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots + ( - 1 ) ^ { n - 1 } n ^ { 2 } = ( - 1 ) ^ { n - 1 } \frac { n ( n + 1 ) } { 2 }\).

7 Fig. 7 shows a sketch of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes. Hence write down the solution of the inequality \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } > 0\).
2. The equation \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } = k\) has no real solutions. Show that \(- 1 < k < - \frac { 1 } { 9 }\). Relate this result to the graph of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\).

8

1. Indicate on an Argand diagram the set of points \(z\) for which \(| z - ( - 8 + 15 \mathrm { j } ) | < 10\).
2. Using the diagram, show that \(7 < | z | < 27\).
3. Mark on your Argand diagram the point, \(P\), at which \(| z - ( - 8 + 15 \mathrm { j } ) | = 10\) and \(\arg z\) takes its maximum value. Find the modulus and argument of \(z\) at \(P\).

9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 8 & - 7 & - 12
- 10 & 5 & 15
- 9 & 6 & 6 \end{array} \right)\) and \(\mathbf { A } ^ { - 1 } = k \left( \begin{array} { r r r } 4 & 2 & 3
5 & 4 & 0
1 & - 1 & 2 \end{array} \right)\).

1. Find the exact value of \(k\).
2. Using your answer to part (i), solve the following simultaneous equations. $$\begin{aligned} 8 x - 7 y - 12 z & = 14
   - 10 x + 5 y + 15 z & = - 25
   - 9 x + 6 y + 6 z & = 3 \end{aligned}$$ You are also given that \(\mathbf { B } = \left( \begin{array} { r r r } - 7 & 5 & 15
   a & - 8 & - 21
   2 & - 1 & - 3 \end{array} \right)\) and \(\mathbf { B } ^ { - 1 } = \frac { 1 } { 3 } \left( \begin{array} { r r r } 1 & 0 & 5
   - 4 & - 3 & 1
   2 & 1 & b \end{array} \right)\).
3. Find the values of \(a\) and \(b\).
4. Write down an expression for \(( \mathbf { A B } ) ^ { - 1 }\) in terms of \(\mathbf { A } ^ { - 1 }\) and \(\mathbf { B } ^ { - 1 }\). Hence find \(( \mathbf { A B } ) ^ { - 1 }\).