OCR MEI FP1 (Further Pure Mathematics 1) 2011 June

1

1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix for a reflection in the line \(y = x\).
3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
4. What single transformation is equivalent to this composite transformation?

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

1. Express \(\frac { z + w } { w }\) in the form \(a + b \mathrm { j }\).
2. Express \(w\) in modulus-argument form.
3. Show \(w\) on an Argand diagram, indicating its modulus and argument.

3 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{gathered} \alpha + \beta + \gamma = 4
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6 \end{gathered}$$ Find \(p\) and \(q\).

4 Solve the inequality \(\frac { 5 x } { x ^ { 2 } + 4 } < x\).

5 Given that \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } \equiv \frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 }\), find \(\sum _ { r = 1 } ^ { 20 } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer as an exact fraction.

6 Prove by induction that \(1 + 8 + 27 + \ldots + n ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\). Section B (36 marks)

7 A curve has equation \(y = \frac { ( x + 9 ) ( 3 x - 8 ) } { x ^ { 2 } - 4 }\).

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.

8 A polynomial \(\mathrm { P } ( z )\) has real coefficients. Two of the roots of \(\mathrm { P } ( z ) = 0\) are \(2 - \mathrm { j }\) and \(- 1 + 2 \mathrm { j }\).

1. Explain why \(\mathrm { P } ( z )\) cannot be a cubic. You are given that \(\mathrm { P } ( z )\) is a quartic.
2. Write down the other roots of \(\mathrm { P } ( z ) = 0\) and hence find \(\mathrm { P } ( z )\) in the form \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d\).
3. Show the roots of \(\mathrm { P } ( z ) = 0\) on an Argand diagram and give, in terms of \(z\), the equation of the circle they lie on.

9 The simultaneous equations $$\begin{aligned} & 2 x - y = 1
& 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).

1. Write down the matrix \(\mathbf { M }\).
2. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists. Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
3. What can you say about the solutions of the equations when \(k = - \frac { 3 } { 2 }\) ?
4. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
   (A) when \(k = 5\) and \(b = 21\),
   (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
   (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\). RECOGNISING ACHIEVEMENT