OCR MEI FP1 (Further Pure Mathematics 1) 2006 January

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

1. Calculate, where possible, \(2 \mathbf { B } , \mathbf { A } + \mathbf { C } , \mathbf { C A }\) and \(\mathbf { A } - \mathbf { B }\).
2. Show that matrix multiplication is not commutative.

2

1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).

3 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 )\), expressing your answer in a fully factorised form.

4 The matrix equation \(\left( \begin{array} { r r } 6 & - 2
- 3 & 1 \end{array} \right) \binom { x } { y } = \binom { a } { b }\) represents two simultaneous linear equations in \(x\) and \(y\).

1. Write down the two equations.
2. Evaluate the determinant of \(\left( \begin{array} { r r } 6 & - 2
   - 3 & 1 \end{array} \right)\). What does this value tell you about the solution of the equations in part (i)?

5 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Find the cubic equation with roots \(2 \alpha , 2 \beta\) and \(2 \gamma\), simplifying your answer as far as possible.

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

7 A curve has equation \(y = \frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } }\).

1. Show that \(y\) can never be zero.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
4. Sketch the curve.
5. Solve the inequality \(\frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } } \leqslant - 2\).

8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
2. Find the other two roots of the equation.
3. Represent the three roots on an Argand diagram.

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 \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point \(( 10,50 )\) under transformation T .
2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
6. Find the \(2 \times 2\) matrix which represents the transformation.
7. Show that this matrix is singular. Relate this result to the transformation.