OCR FP1 (Further Pure Mathematics 1) 2010 June

Prove by induction that, for \(n \geq 1\), \(\sum_{r=1}^{n} r(r + 1) = \frac{1}{3}n(n + 1)(n + 2)\). [5]

The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & -4 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 3 & 0 \\ -2 & 2 \end{pmatrix}\). Find

1. \(\mathbf{AB}\), [2]
2. \(\mathbf{BA} - 4\mathbf{C}\). [4]

Find \(\sum_{r=1}^{n} (2r - 1)^2\), expressing your answer in a fully factorised form. [6]

The complex numbers \(a\) and \(b\) are given by \(a = 7 + 6\text{i}\) and \(b = 1 - 3\text{i}\). Showing clearly how you obtain your answers, find

1. \(|a - 2b|\) and \(\arg(a - 2b)\), [4]
2. \(\frac{b}{a}\), giving your answer in the form \(x + \text{i}y\). [3]

1. Write down the matrix that represents a reflection in the line \(y = x\). [2]
2. Describe fully the geometrical transformation represented by each of the following matrices:
   1. \(\begin{pmatrix} 5 & 0 \\ 0 & 1 \end{pmatrix}\), [2]
   2. \(\begin{pmatrix} \frac{1}{2} & \frac{1}{2}\sqrt{3} \\ -\frac{1}{2}\sqrt{3} & \frac{1}{2} \end{pmatrix}\). [2]

1. Sketch on a single Argand diagram the loci given by
   1. \(|z - 3 + 4\text{i}| = 5\), [2]
   2. \(|z| = |z - 6|\). [2]
2. Indicate, by shading, the region of the Argand diagram for which $$|z - 3 + 4\text{i}| \leq 5 \quad \text{and} \quad |z| \geq |z - 6|.$$ [2]

The quadratic equation \(x^2 + 2kx + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac{\alpha + \beta}{\alpha}\) and \(\frac{\alpha + \beta}{\beta}\). [7]

1. Show that \(\frac{1}{\sqrt{r + 2} + \sqrt{r}} = \frac{\sqrt{r + 2} - \sqrt{r}}{2}\). [2]
2. Hence find an expression, in terms of \(n\), for $$\sum_{r=1}^{n} \frac{1}{\sqrt{r + 2} + \sqrt{r}}.$$ [6]
3. State, giving a brief reason, whether the series \(\sum_{r=1}^{\infty} \frac{1}{\sqrt{r + 2} + \sqrt{r}}\) converges. [1]

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

1. Find, in terms of \(a\), the determinant of \(\mathbf{A}\). [3]
2. Three simultaneous equations are shown below. \begin{align} ax + ay - z &= -1
   ay + 2z &= 2a
   x + 2y + z &= 1 \end{align} For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
   1. \(a = 0\)
   2. \(a = 1\)
   3. \(a = 2\) [6]

The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).

1. Use an algebraic method to find \(z\). [5]
2. Show that \(z^3 = 2 + 11\text{i}\). [1]

The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).

1. Find \(w\). [5]