OCR FP1 (Further Pure Mathematics 1) 2005 June

The complex numbers \(2 + 3\text{i}\) and \(4 - \text{i}\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \text{i}y\), showing clearly how you obtain your answers.

1. \(z + 5w\), [2]
2. \(z^*w\), where \(z^*\) is the complex conjugate of \(z\), [3]
3. \(\frac{1}{w}\). [2]

Use an algebraic method to find the square roots of the complex number \(21 - 20\text{i}\). [6]

1. Show that $$\frac{r+1}{r+2} - \frac{r}{r+1} = \frac{1}{(r+1)(r+2)}.$$ [2]
2. Hence find an expression, in terms of \(n\), for $$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots + \frac{1}{(n+1)(n+2)}.$$ [4]
3. Hence write down the value of \(\sum_{r=1}^{\infty} \frac{1}{(r+1)(r+2)}\). [1]

The loci \(C_1\) and \(C_2\) are given by $$|z - 2\text{i}| = 2 \quad \text{and} \quad |z + 1| = |z + \text{i}|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} a & 1 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}\).

1. Given that \(\mathbf{B}\) is singular, show that \(a = -\frac{2}{3}\). [3]
2. Given instead that \(\mathbf{B}\) is non-singular, find the inverse matrix \(\mathbf{B}^{-1}\). [4]
3. Hence, or otherwise, solve the equations \begin{align} -x + y + 3z &= 1,
   2x + y - z &= 4,
   y + 2z &= -1. \end{align} [3]

1. Write down the matrix \(\mathbf{C}\) which represents a stretch, scale factor 2, in the \(x\)-direction. [2]
2. The matrix \(\mathbf{D}\) is given by \(\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}\). Describe fully the geometrical transformation represented by \(\mathbf{D}\). [2]
3. The matrix \(\mathbf{M}\) represents the combined effect of the transformation represented by \(\mathbf{C}\) followed by the transformation represented by \(\mathbf{D}\). Show that $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
4. Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]