OCR MEI FP1 (Further Pure Mathematics 1) 2013 June

1 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x \left( x ^ { 2 } - 5 \right) \equiv ( x - 2 ) \left( A x ^ { 2 } + B x + C \right) + D\).

2 You are given that \(z = \frac { 3 } { 2 }\) is a root of the cubic equation \(2 z ^ { 3 } + 9 z ^ { 2 } + 2 z - 30 = 0\). Find the other two roots.

3 You are given that \(\mathbf { N } = \left( \begin{array} { r r r } - 9 & - 2 & - 4
3 & 2 & 2
5 & 1 & 2 \end{array} \right)\) and \(\mathbf { N } ^ { - 1 } = \left( \begin{array} { r r r } 1 & 0 & 2
2 & 1 & 3
- \frac { 7 } { 2 } & p & - 6 \end{array} \right)\).

1. Find the value of \(p\).
2. Solve the equation \(\mathbf { N } \left( \begin{array} { c } x
   y
   z \end{array} \right) = \left( \begin{array} { r } - 39
   5
   22 \end{array} \right)\).

4 The complex number \(z _ { 1 }\) is \(3 - 2 \mathrm { j }\) and the complex number \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 4 }\).

1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), giving \(a\) and \(b\) in exact form.
2. Represent \(z _ { 1 } , z _ { 2 } , z _ { 1 } + z _ { 2 }\) and \(z _ { 1 } - z _ { 2 }\) on a single Argand diagram.

5 You are given that \(\frac { 4 } { ( 4 n - 3 ) ( 4 n + 1 ) } \equiv \frac { 1 } { 4 n - 3 } - \frac { 1 } { 4 n + 1 }\). Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 4 r - 3 ) ( 4 r + 1 ) } = \frac { n } { 4 n + 1 }$$

6 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + 3 x - 6 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with roots \(\frac { \alpha } { 3 } + 1 , \frac { \beta } { 3 } + 1\) and \(\frac { \gamma } { 3 } + 1\), simplifying your answer as far as possible.

7 Fig. 7 shows an incomplete sketch of \(y = \frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) }\) where \(a , b\) and \(c\) are integers. The asymptotes of the curve are also shown. \begin{figure}[h] \end{figure}

1. Determine the values of \(a , b\) and \(c\). Use these values of \(a , b\) and \(c\) throughout the rest of the question.
2. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\), justifying your answer. On the copy of Fig. 7, sketch the rest of the curve.
3. Find the \(x\) coordinates of the points on the curve where \(y = 1\). Write down the solution to the inequality \(\frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) } < 1\).

8

1. Use standard series formulae to show that $$\sum _ { r = 1 } ^ { n } [ r ( r - 1 ) - 1 ] = \frac { 1 } { 3 } n ( n + 2 ) ( n - 2 )$$
2. Prove (*) by mathematical induction.

9

1. Describe fully the transformation Q , represented by the matrix \(\mathbf { Q }\), where \(\mathbf { Q } = \left( \begin{array} { r l } 0 & 1
   - 1 & 0 \end{array} \right)\). The transformation M is represented by the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { r r } 0 & - 1
   0 & 1 \end{array} \right)\).
2. M maps all points on the line \(y = 2\) onto a single point, P. Find the coordinates of P.
3. M maps all points on the plane onto a single line, \(l\). Find the equation of \(l\).
4. M maps all points on the line \(n\) onto the point ( - 6 , 6). Find the equation of \(n\).
5. Show that \(\mathbf { M }\) is singular. Relate this to the transformation it represents.
6. R is the composite transformation M followed by Q . R maps all points on the plane onto the line \(q\). Find the equation of \(q\).