OCR MEI FP1 (Further Pure Mathematics 1)

1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3 \\ - 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?

2

1. Show that \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
2. Hence use the method of differences to find the sum of the series $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$

3

1. Solve the equation \(\frac { 1 } { x + 2 } = 3 x + 4\).
2. Solve the inequality \(\frac { 1 } { x + 2 } \leqslant 3 x + 4\).

4 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 )\), giving your answer in a factorised form.

5 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), simplifying your answer as far as you can.

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

7 A curve has equation \(y = \frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) }\).

1. Write down the values of \(x\) for which \(y = 0\).
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or from below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.
5. Solve the inequality \(\frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) } \leqslant 2\).

8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).

1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).

9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
2. Explain how your answer to part (i) relates to this information.
3. By investigating the invariant points of the reflection, find the equation of the mirror line.
4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS
   4755
   \textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}

   Tuesday 7 JUNE 2005	Afternoon	1 hour 30 minutes
   Additional materials: Answer booklet Graph paper MEI Examination Formulae and Tables (MF2)

   TIME 1 hour 30 minutes
   • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
   • Answer all the questions.
   • You are permitted to use a graphical calculator in this paper.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • You are advised that an answer may receive no marks unless you show sufficient defail of the working to indicate that a correct method is being used.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The total number of marks for this paper is 72.

10

1. You are given that $$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$ Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
2. Hence find the sum of the infinite series $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$ RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS} Further Concepts For Advanced Mathematics (FP1)
   Wednesday 18 JANUARY 2006 Afternoon ..... 1 hour 30 minutes
   Additional materials:
   8 page answer booklet
   Graph paper
   MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes
   • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
   • Answer all the questions.
   • You are permitted to use a graphical calculator in this paper.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
   • The total number of marks for this paper is 72.