SPS SPS FM (SPS FM) 2023 February

Matrices A and B are given by \(\mathbf{A} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}\). Use A and B to disprove the proposition: "Matrix multiplication is commutative". [2]

A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]

Express \(\frac{(x-7)(x-2)}{(x+2)(x-1)^2}\) in partial fractions. [5]

1. You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of k. [4]
2. Determine whether the line with equation \(y = kx\) in part above is a line of invariant points under T. [1]

1. Expand \(\sqrt{1 + 2x}\) in ascending powers of x, up to and including the term in \(x^3\). [4]
2. Hence expand \(\frac{\sqrt{1 + 2x}}{1 + 9x^2}\) in ascending powers of x, up to and including the term in \(x^3\). [3]
3. Determine the range of values of x for which the expansion in part (b) is valid. [2]

1. The members of a team stand in a random order in a straight line for a photograph. There are four men and six women. Find the probability that all the men are next to each other. [3]
2. Find the probability that no two men are next to one another. [4]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ P is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of P. [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

Q is a point on \(l_1\) which is 12 metres away from P. R is the point on \(l_2\) such that QR is perpendicular to \(l_1\).

1. Determine the length QR. [2]

In this question you must show detailed reasoning. The equation f(x) = 0, where f(x) = \(x^4 + 2x^3 + 2x^2 + 26x + 169\), has a root x = 2 + 3i.

1. Express f(x) as a product of two quadratic factors. [4]
2. Hence write down all the roots of the equation f(x) = 0. [1]

O is the origin of a coordinate system whose units are cm. The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively. The arc BC is part of the curve with equation \(x^2 + (y - 10)^2 = 37\). The closed shape OABCD is formed, in turn, from the line segments OA and AB, the arc BC and the line segments CD and DO (see diagram). A funnel can be modelled by rotating OABCD by \(2\pi\) radians about the y-axis. \includegraphics{figure_9} Find the volume of the funnel according to the model. [3]

A transformation is equivalent to a shear parallel to the x-axis followed by a shear parallel to the y-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of s the matrices which represent each of the shears. [7]

Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and Re (z) \(\geq\) 9. [6]