OCR MEI FP1 (Further Pure Mathematics 1) 2007 June

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

1. Find the inverse of \(\mathbf{M}\). [2]
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. [1]

Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}

Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]

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]
2. Express \(\alpha\beta\) in the form \(a + b\mathrm{j}\). [2]
3. Express \(\frac{\alpha + \beta}{\beta}\) in the form \(a + b\mathrm{j}\). [3]

The roots of the cubic equation \(x^3 + 3x^2 - 7x + 1 = 0\) 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. [6]

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

Prove by induction that \(\sum_{r=1}^{n} 3^{r-1} = \frac{3^n - 1}{2}\). [6]

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. [3]
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote. [4]
3. Determine whether the curve approaches the horizontal asymptote from above or below for
   1. large positive values of \(x\),
   2. large negative values of \(x\). [3]
4. Sketch the curve. [4]

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

1. Write down the other complex root. [1]
2. Explain why the equation must have a real root. [1]
3. Find the value of the real root and the values of \(A\) and \(B\). [9]

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

1. Find the value of \(n\). [2]
2. Write down the inverse matrix \(\mathbf{A}^{-1}\) and state the condition on \(k\) for this inverse to exist. [4]
3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. \begin{align} x - 2y + z &= 1
   2x + y + 2z &= 12
   3x + 2y - z &= 3 \end{align} [5]