OCR MEI FP1 (Further Pure Mathematics 1) 2012 June

1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right)\) represents a transformation B .

1. Describe the transformations A and B .
2. Find the matrix representing the composite transformation consisting of A followed by B .
3. What single transformation is represented by this matrix?

2 You are given that \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers.
\(z _ { 1 } = 3 + 3 \sqrt { 3 } \mathrm { j }\), and \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 3 }\).

1. Find the modulus and argument of \(z _ { 1 }\), giving your answers exactly.
2. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), where \(a\) and \(b\) are to be given exactly.
3. Explain why, when plotted on an Argand diagram, \(z _ { 1 } , z _ { 2 }\) and the origin lie on a straight line.

3 The cubic equation \(3 x ^ { 3 } + 8 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { \alpha } { 6 }\) and \(\alpha - 7\). Find the values of \(\alpha , p\) and \(q\).

4 Solve the inequality \(\frac { 3 } { x - 4 } > 1\).

5

1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { 30 } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), expressing your answer as a fraction.

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

1. Calculate the value of the third term, \(a _ { 3 }\).
2. Prove by induction that \(a _ { n } = \frac { 5 \times 3 ^ { n - 1 } - 3 } { 2 }\).

7 A curve has equation \(y = \frac { x ^ { 2 } - 25 } { ( x - 3 ) ( x + 4 ) ( 3 x + 2 ) }\).

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the asymptotes.
3. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\).
4. Sketch the curve.

8

1. Verify that \(1 + 3 \mathrm { j }\) is a root of the equation \(3 z ^ { 3 } - 2 z ^ { 2 } + 22 z + 40 = 0\), showing your working.
2. Explain why the equation must have exactly one real root.
3. Find the other roots of the equation.

9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 3 & - 4 & 1
2 & 1 & k
7 & - 1 & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r c } - 4 & - 5 & 11
- 19 & - 4 & - 7
- 9 & - 31 & 2 - k \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } 79 & 0 & - 3 - k
- 9 k - 27 & - 31 k - 14 & q
p & 0 & 82 + k \end{array} \right)\) where \(p\) and \(q\) are to be determined.

1. Show that \(p = 0\) and \(q = 15 + 2 k - k ^ { 2 }\). It is now given that \(k = - 3\).
2. Find \(\mathbf { A B }\) and hence write down the inverse matrix \(\mathbf { A } ^ { - 1 }\).
3. Use a matrix method to find the values of \(x , y\) and \(z\) that satisfy the equation \(\mathbf { A } \left( \begin{array} { l } x
   y
   z \end{array} \right) = \left( \begin{array} { r } 14
   - 23
   9 \end{array} \right)\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}