OCR MEI FP1 (Further Pure Mathematics 1) 2007 January

1 Is the following statement true or false? Justify your answer. $$x ^ { 2 } = 4 \text { if and only if } x = 2$$

2

1. Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 7 = 0\), simplifying your answers as far as possible.
2. Represent these roots on an Argand diagram.

3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0
0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h] \end{figure}

1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).

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

5 The roots of the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(2 \alpha + 1,2 \beta + 1\) and \(2 \gamma + 1\), expressing your answer in a form with integer coefficients.

6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\).

7 A curve has equation \(y = \frac { 5 } { ( x + 2 ) ( 4 - x ) }\).

1. Write down the value of \(y\) when \(x = 0\).
2. Write down the equations of the three asymptotes.
3. Sketch the curve.
4. Find the values of \(x\) for which \(\frac { 5 } { ( x + 2 ) ( 4 - x ) } = 1\) and hence solve the inequality $$\frac { 5 } { ( x + 2 ) ( 4 - x ) } < 1 .$$

8 It is given that \(m = - 4 + 2 \mathrm { j }\).

1. Express \(\frac { 1 } { m }\) in the form \(a + b \mathrm { j }\).
2. Express \(m\) in modulus-argument form.
3. Represent the following loci on separate Argand diagrams.
   (A) \(\arg ( z - m ) = \frac { \pi } { 4 }\)
   (B) \(0 < \arg ( z - m ) < \frac { \pi } { 4 }\)

9 Matrices \(\mathbf { M }\) and \(\mathbf { N }\) are given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 2
0 & 1 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { r r } 1 & - 3
1 & 4 \end{array} \right)\).

1. Find \(\mathbf { M } ^ { - 1 }\) and \(\mathbf { N } ^ { - 1 }\).
2. Find \(\mathbf { M N }\) and \(( \mathbf { M N } ) ^ { - \mathbf { 1 } }\). Verify that \(( \mathbf { M N } ) ^ { - 1 } = \mathbf { N } ^ { - 1 } \mathbf { M } ^ { - 1 }\).
3. The result \(( \mathbf { P Q } ) ^ { - 1 } = \mathbf { Q } ^ { - 1 } \mathbf { P } ^ { - 1 }\) is true for any two \(2 \times 2\), non-singular matrices \(\mathbf { P }\) and \(\mathbf { Q }\). The first two lines of a proof of this general result are given below. Beginning with these two lines, complete the general proof. $$\begin{aligned} & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q } = \mathbf { I }
   \Rightarrow & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q Q } \mathbf { Q } ^ { - 1 } = \mathbf { I Q } ^ { - 1 } \end{aligned}$$