OCR FP1 (Further Pure Mathematics 1)

Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]

The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.

1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]

The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), 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 \(21 - 20i\). [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 $$|z - 2i| = 2 \quad \text{and} \quad |z + 1| = |z + i|$$ 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. The quadratic equation \(x^2 - 2x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
   1. Write down the values of \(\alpha + \beta\) and \(\alpha\beta\). [2]
   2. Show that \(\alpha^2 + \beta^2 = -4\). [2]
   3. Hence find a quadratic equation which has roots \(\alpha^2\) and \(\beta^2\). [3]
2. The cubic equation \(x^3 - 12x^2 + ax - 48 = 0\) has roots \(p\), \(2p\) and \(3p\).
   1. Find the value of \(p\). [2]
   2. Hence find the value of \(a\). [2]

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]