OCR MEI FP1 (Further Pure Mathematics 1) 2008 January

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

1. Find BA.
2. A plane shape of area 3 square units is transformed using matrix \(\mathbf { A }\). The image is transformed using matrix B. What is the area of the resulting shape?

2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).

1. Calculate \(\alpha ^ { 2 }\).
2. Express \(\alpha\) in modulus-argument form.

3

1. Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.
2. Show the roots on an Argand diagram.

4 Using the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { n } [ ( r + 1 ) ( r - 2 ) ] = \frac { 1 } { 3 } n \left( n ^ { 2 } - 7 \right)\).

5 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{aligned} \alpha + \beta + \gamma & = 3
\alpha \beta \gamma & = - 7
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 13 \end{aligned}$$

1. Write down the values of \(p\) and \(r\).
2. Find the value of \(q\).

6 A sequence is defined by \(a _ { 1 } = 7\) and \(a _ { k + 1 } = 7 a _ { k } - 3\).

1. Calculate the value of the third term, \(a _ { 3 }\).
2. Prove by induction that \(a _ { n } = \frac { \left( 13 \times 7 ^ { n - 1 } \right) + 1 } { 2 }\).

7 The sketch below shows part of the graph of \(y = \frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) }\). One section of the graph has been omitted. \begin{figure}[h] \end{figure}

1. Find the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
3. Copy the sketch and draw in the missing section.
4. Solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) } \geqslant 0\).

8

1. On a single Argand diagram, sketch the locus of points for which
   (A) \(| z - 3 \mathrm { j } | = 2\),
   (B) \(\quad \arg ( z + 1 ) = \frac { 1 } { 4 } \pi\).
2. Indicate clearly on your Argand diagram the set of points for which $$| z - 3 \mathrm { j } | \leqslant 2 \quad \text { and } \quad \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi .$$
3. (A) By drawing an appropriate line through the origin, indicate on your Argand diagram the point for which \(| z - 3 j | = 2\) and \(\arg z\) has its minimum possible value.
   (B) Calculate the value of \(\arg z\) at this point.

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = x\) and has the same \(x\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point ( \(- 3,7\) ) under transformation T .
2. Write down the image of the point \(( x , y )\) under transformation T .
3. Find the \(2 \times 2\) matrix which represents the transformation.
4. Describe the transformation M represented by the matrix \(\left( \begin{array} { r r } 0 & - 1
   1 & 0 \end{array} \right)\).
5. Find the matrix representing the composite transformation of T followed by M .
6. Find the image of the point \(( x , y )\) under this composite transformation. State the equation of the line on which all of these images lie.