SPS SPS FM Pure (SPS FM Pure) 2025 January

\includegraphics{figure_1} The diagram shows the curve \(y = 6x - x^2\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning. [4]

1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) is constant. A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).

1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [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. 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]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]

The point \(A\), with position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\), lies on \(l_1\) The point \(B\) is the image of \(A\) after reflection in the line \(l_2\)

1. Find the position vector of \(B\). [3]

You are given the complex number \(w = 2 + 2\sqrt{3}i\).

1. Express \(w\) in modulus-argument form. [3]
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]

A candlestick has base diameter \(8\) cm and height \(28\) cm, as shown in Figure \(9\). A model of the candlestick is shown in Figure \(10\), together with the equations that were used to create the model. \includegraphics{figure_7}

1. Show that the volume generated by rotating the shaded region (in Figure \(10\)) \(2\pi\) radians about the \(y\)-axis is \(\frac{112}{15}\pi\). [4]
2. Hence find the volume of metal needed to create the candlestick. [4]