OCR MEI FP1 (Further Pure Mathematics 1) 2007 June

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

1. Find the inverse of \(\mathbf { M }\).
2. A triangle of area 2 square units undergoes the transformation represented by the matrix \(\mathbf { M }\). Find the area of the image of the triangle following this transformation.

2 Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. \begin{figure}[h] \end{figure}

3 Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x ^ { 3 } - 4 \equiv ( x - 1 ) \left( A x ^ { 2 } + B x + C \right) + D .$$

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

1. Represent \(\beta\) and its complex conjugate \(\beta ^ { * }\) on an Argand diagram.
2. Express \(\alpha \beta\) in the form \(a + b \mathrm { j }\).
3. Express \(\frac { \alpha + \beta } { \beta }\) in the form \(a + b \mathrm { j }\).

5 The roots of the cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha , 3 \beta\) and \(3 \gamma\), expressing your answer in a form with integer coefficients.

1. Show that \(\frac { 1 } { r + 2 } - \frac { 1 } { r + 3 } = \frac { 1 } { ( r + 2 ) ( r + 3 ) }\).
2. Hence use the method of differences to find \(\frac { 1 } { 3 \times 4 } + \frac { 1 } { 4 \times 5 } + \frac { 1 } { 5 \times 6 } + \ldots + \frac { 1 } { 52 \times 53 }\).

7 Prove by induction that \(\sum _ { r = 1 } ^ { n } 3 ^ { r - 1 } = \frac { 3 ^ { n } - 1 } { 2 }\).

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

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
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.

9 The cubic equation \(x ^ { 3 } + A x ^ { 2 } + B x + 15 = 0\), where \(A\) and \(B\) are real numbers, has a root \(x = 1 + 2 \mathrm { j }\).

1. Write down the other complex root.
2. Explain why the equation must have a real root.
3. Find the value of the real root and the values of \(A\) and \(B\).

10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & k
2 & 1 & 2
3 & 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r c c } - 5 & - 2 + 2 k & - 4 - k
8 & - 1 - 3 k & - 2 + 2 k
1 & - 8 & 5 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } k - n & 0 & 0
0 & k - n & 0
0 & 0 & k - n \end{array} \right)\).

1. Find the value of \(n\).
2. Write down the inverse matrix \(\mathbf { A } ^ { - 1 }\) and state the condition on \(k\) for this inverse to exist.
3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. $$\begin{aligned} x - 2 y + z = & 1
   2 x + y + 2 z = & 12
   3 x + 2 y - z = & 3 \end{aligned}$$