Edexcel FP1 (Further Pure Mathematics 1)

$$\text{f}(x) = 2x^3 - 8x^2 + 7x - 3.$$ Given that \(x = 3\) is a solution of the equation f\((x) = 0\), solve f\((x) = 0\) completely. [5]

1. Show, using the formulae for \(\sum r\) and \(\sum r^2\), that $$\sum_{r=1}^n (6r^2 + 4r - 1) = n(n + 2)(2n + 1).$$ [5]
2. Hence, or otherwise, find the value of \(\sum_{r=1}^n (6r^2 + 4r - 1)\). [2]

The rectangular hyperbola, \(H\), has parametric equations \(x = 5t, y = \frac{5}{t}, t \neq 0\).

1. Write the cartesian equation of \(H\) in the form \(xy = c^2\). [1]
2. Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively. Find the coordinates of the mid-point of \(AB\). [3]

Prove by induction that, for \(n \in \mathbb{Z}^+\), $$\sum_{r=1}^n \frac{1}{r(r+1)} = \frac{n}{n+1}$$ [5]

$$\text{f}(x) = 3\sqrt{x} + \frac{18}{\sqrt{x}} - 20.$$

1. Show that the equation f\((x) = 0\) has a root \(a\) in the interval \([1.1, 1.2]\). [2]
2. Find \(f'(x)\). [3]
3. Using \(x_0 = 1.1\) as a first approximation to \(a\), apply the Newton-Raphson procedure once to f\((x)\) to find a second approximation to \(a\), giving your answer to 3 significant figures. [4]

A series of positive integers \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 6 \text{ and } u_{n+1} = 6u_n - 5, \text{ for } n \geq 1.$$ Prove by induction that \(u_n = 5 \times 6^{n-1} + 1\), for \(n \geq 1\). [5]

Given that \(\mathbf{X} = \begin{pmatrix} 2 & a \\ -1 & -1 \end{pmatrix}\), where \(a\) is a constant, and \(a \neq 2\).

1. find \(\mathbf{X}^{-1}\) in terms of \(a\). [3]
2. Given that \(\mathbf{X} + \mathbf{X}^{-1} = \mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix, find the value of \(a\). [3]

A parabola has equation \(y^2 = 4ax\), \(a > 0\). The point \(Q (aq^2, 2aq)\) lies on the parabola.

1. Show that an equation of the tangent to the parabola at \(Q\) is $$yq = x + aq^2.$$ [4]
2. This tangent meets the \(y\)-axis at the point \(R\). Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\). [3]
3. Show that \(l\) passes through the focus of the parabola. [1]
4. Find the coordinates of the point where \(l\) meets the directrix of the parabola. [2]

Given that \(z_1 = 3 + 2i\) and \(z_2 = \frac{12 - 5i}{z_1}\).

1. Find \(z_2\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2]
2. Show, on an Argand diagram, the point \(P\) representing \(z_1\) and the point \(Q\) representing \(z_2\). [2]
3. Given that \(O\) is the origin, show that \(\angle POQ = \frac{\pi}{2}\). [2]

The circle passing through the points \(O\), \(P\) and \(Q\) has centre \(C\). Find

1. the complex number represented by \(C\), [2]
2. the exact value of the radius of the circle. [2]

$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

1. Find \(\mathbf{D}\). [2]
2. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]

The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).

1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]