OCR FP1 (Further Pure Mathematics 1) 2013 January

The matrix A is given by \(A = \begin{pmatrix} a & 1 \\ 1 & a \end{pmatrix}\), where \(a \neq \frac{1}{2}\), and I denotes the \(2 \times 2\) identity matrix. Find

1. \(2A - 3I\), [3]
2. \(A^{-1}\). [2]

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

The complex number \(2 - i\) is denoted by \(z\).

1. Find \(|z|\) and \(\arg z\). [2]
2. Given that \(az + bz^* = 4 - 8i\), find the values of the real constants \(a\) and \(b\). [5]

The quadratic equation \(x^2 + x + k = 0\) has roots \(\alpha\) and \(\beta\).

1. Use the substitution \(x = 2u + 1\) to obtain a quadratic equation in \(u\). [2]
2. Hence, or otherwise, find the value of \(\left(\frac{\alpha - 1}{2}\right)\left(\frac{\beta - 1}{2}\right)\) in terms of \(k\). [2]

By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations \begin{align} 3x + 2y + 4z &= 5,
\lambda y + z &= 1,
x + \lambda y + \lambda z &= 4, \end{align} do not have a unique solution for \(x\), \(y\) and \(z\). [6]

\includegraphics{figure_6} The diagram shows the unit square \(OABC\), and its image \(OA'B'C'\) after a transformation. The points have the following coordinates: \(A(1, 0)\), \(B(1, 1)\), \(C(0, 1)\), \(B'(3, 2)\) and \(C'(2, 2)\).

1. Write down the matrix, X, for this transformation. [2]
2. The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
3. Find the matrix that represents transformation Q followed by transformation P. [2]

1. Sketch on a single Argand diagram the loci given by
   1. \(|z| = 2\), [2]
   2. \(\arg(z - 3 - i) = \pi\). [3]
2. Indicate, by shading, the region of the Argand diagram for which $$|z| < 2 \text{ and } 0 < \arg(z - 3 - i) < \pi.$$ [2]

1. Show that \(\frac{1}{r} - \frac{3}{r+1} + \frac{2}{r+2} = \frac{2-r}{r(r+1)(r+2)}\). [2]
2. Hence show that \(\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = -\frac{n}{(n+1)(n+2)}\). [5]
3. Find the value of \(\sum_{r=3}^{\infty} \frac{2-r}{r(r+1)(r+2)}\). [2]

1. Show that \((\alpha\beta + \beta\gamma + \gamma\alpha)^2 = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\alpha\beta\gamma(\alpha + \beta + \gamma)\). [3]
2. It is given that \(\alpha\), \(\beta\) and \(\gamma\) are the roots of the cubic equation \(x^3 + px^2 - 4x + 3 = 0\), where \(p\) is a constant. Find the value of \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}\) in terms of \(p\). [5]

The sequence \(u_1, u_2, u_3, \ldots\) is defined by \(u_1 = 2\) and \(u_{n+1} = \frac{u_n}{1 + u_n}\) for \(n \geq 1\).

1. Find \(u_2\) and \(u_3\), and show that \(u_4 = \frac{2}{7}\). [3]
2. Hence suggest an expression for \(u_n\). [2]
3. Use induction to prove that your answer to part (ii) is correct. [5]