OCR MEI FP1 (Further Pure Mathematics 1) 2011 January

1 Find the values of \(P , Q , R\) and \(S\) in the identity \(3 x ^ { 3 } + 18 x ^ { 2 } + P x + 31 \equiv Q ( x + R ) ^ { 3 } + S\).

2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 4 & 0 \\ - 1 & 3 \end{array} \right)\).

1. The transformation associated with \(\mathbf { M }\) is applied to a figure of area 3 square units. Find the area of the transformed figure.
2. Find \(\mathbf { M } ^ { - 1 }\) and \(\operatorname { det } \mathbf { M } ^ { - 1 }\).
3. Explain the significance of \(\operatorname { det } \mathbf { M } \times \operatorname { det } \mathbf { M } ^ { - 1 }\) in terms of transformations.

3 The roots of the cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).

4 Represent on an Argand diagram the region defined by \(2 < | z - ( 3 + 2 \mathrm { j } ) | \leqslant 3\).

5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 3 - 4 r ) = \frac { 1 } { 2 } n ( n + 1 ) \left( 1 - 2 n ^ { 2 } \right)\).

6 A sequence is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 ^ { n + 1 }\). Prove by induction that \(u _ { n } = 2 ^ { n + 1 } + 1\).

7 Fig. 7 shows part of the curve with equation \(y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the coordinates of the two points where the curve crosses the axes.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Determine how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
4. On the copy of Fig. 7, sketch the rest of the curve.
5. Solve the inequality \(\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0\).

8 The function \(\mathrm { f } ( z ) = z ^ { 4 } - z ^ { 3 } + a z ^ { 2 } + b z + c\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha , \beta\), \(\gamma\) and \(\delta\) where \(\alpha = 1\) and \(\beta = 1 + \mathrm { j }\).

1. Write down the other complex root and explain why the equation must have a second real root.
2. Write down the value of \(\alpha + \beta + \gamma + \delta\) and find the second real root.
3. Find the values of \(a , b\) and \(c\).
4. Write down \(\mathrm { f } ( - z )\) and the roots of \(\mathrm { f } ( - z ) = 0\).

\(\mathbf { 9 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 5 \\ 3 & a & 1 \\ 1 & - 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 2 a + 1 & 3 & 1 + 5 a \\ - 5 & 1 & - 13 \\ - 3 - a & - 1 & - 2 a - 3 \end{array} \right)\).

1. Show that \(\mathbf { A B } = ( 8 + a ) \mathbf { I }\).
2. State the value of \(a\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Write down \(\mathbf { A } ^ { - 1 }\) in terms of \(a\), when \(\mathbf { A } ^ { - 1 }\) exists.
3. Use \(\mathbf { A } ^ { - 1 }\) to solve the following simultaneous equations. $$\begin{aligned} - 2 x + y - 5 z & = - 55 \\ 3 x + 4 y + z & = - 9 \\ x - y + 2 z & = 26 \end{aligned}$$
4. What can you say about the solutions of the following simultaneous equations? $$\begin{aligned} - 2 x + y - 5 z & = p \\ 3 x - 8 y + z & = q \\ x - y + 2 z & = r \end{aligned}$$