OCR MEI FP1 (Further Pure Mathematics 1) 2009 June

1

1. Find the inverse of the matrix \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 1
   3 & 2 \end{array} \right)\).
2. Use this inverse to solve the simultaneous equations $$\begin{aligned} & 4 x - y = 49
   & 3 x + 2 y = 100 \end{aligned}$$ showing your working clearly.

2 Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.

3

1. Sketch the graph of \(y = \frac { 2 } { x + 4 }\).
2. Solve the inequality $$\frac { 2 } { x + 4 } \leqslant x + 3$$ showing your working clearly.

4 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } + p x + q = 0\) are \(2 w , - 6 w\) and \(3 w\). Find the values of the roots and the values of \(p\) and \(q\).

5

1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } \equiv \frac { 5 } { ( 5 r - 2 ) ( 5 r + 3 ) }\) for all integers \(r\).
2. Hence use the method of differences to show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }\).

6 Prove by induction that \(3 + 10 + 17 + \ldots + ( 7 n - 4 ) = \frac { 1 } { 2 } n ( 7 n - 1 )\) for all positive integers \(n\). Section B (36 marks)

7 A curve has equation \(y = \frac { ( x + 2 ) ( 3 x - 5 ) } { ( 2 x + 1 ) ( x - 1 ) }\).

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 Fig. 8 shows an Argand diagram. \begin{figure}[h] \end{figure}

1. Write down the equation of the locus represented by the perimeter of the circle in the Argand diagram.
2. Write down the equation of the locus represented by the half-line \(\ell\) in the Argand diagram.
3. Express the complex number represented by the point P in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
4. Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.

9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0
0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1
1 & 0 \end{array} \right)\).

1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN.
   \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.