OCR MEI FP1 (Further Pure Mathematics 1) 2006 June

1. State the transformation represented by the matrix \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\). [1]
2. Write down the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin. [1]
3. Find the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin, followed by reflection in the \(x\)-axis. [2]

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

The cubic equation \(z^3 + 4z^2 - 3z + 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma\), \(\alpha\beta + \beta\gamma + \gamma\alpha\) and \(\alpha\beta\gamma\). [3]
2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 22\). [3]

Indicate, on separate Argand diagrams,

1. the set of points \(z\) for which \(|z-(3-\mathrm{j})| \leqslant 3\), [3]
2. the set of points \(z\) for which \(1 < |z-(3-\mathrm{j})| \leqslant 3\), [2]
3. the set of points \(z\) for which \(\arg(z-(3-\mathrm{j})) = \frac{1}{4}\pi\). [3]

1. The matrix \(\mathbf{S} = \begin{pmatrix} -1 & 2 \\ -3 & 4 \end{pmatrix}\) represents a transformation.
   1. Show that the point \((1, 1)\) is invariant under this transformation. [1]
   2. Calculate \(\mathbf{S}^{-1}\). [2]
   3. Verify that \((1, 1)\) is also invariant under the transformation represented by \(\mathbf{S}^{-1}\). [1]
2. Part (i) may be generalised as follows. If \((x, y)\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf{T}\), it is also invariant under the transformation represented by \(\mathbf{T}^{-1}\). Starting with \(\mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\), or otherwise, prove this result. [2]

Prove by induction that \(3 + 6 + 12 + \ldots + 3 \times 2^{n-1} = 3(2^n - 1)\) for all positive integers \(n\). [7]

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

1. Write down the equations of the three asymptotes. [3]
2. Determine whether the curve approaches the horizontal asymptote from above or from below for
   1. large positive values of \(x\),
   2. large negative values of \(x\). [3]
3. Sketch the curve. [4]
4. Solve the inequality \(\frac{x^2}{(x-2)(x+1)} > 0\). [3]

1. Verify that \(2 + \mathrm{j}\) is a root of the equation \(2x^3 - 11x^2 + 22x - 15 = 0\). [5]
2. Write down the other complex root. [1]
3. Find the third root of the equation. [4]

1. Show that \(r(r+1)(r+2) - (r-1)r(r+1) \equiv 3r(r+1)\). [2]
2. Hence use the method of differences to find an expression for \(\sum_{r=1}^{n} r(r+1)\). [6]
3. Show that you can obtain the same expression for \(\sum_{r=1}^{n} r(r+1)\) using the standard formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\). [5]