SPS SPS FM (SPS FM) 2025 February

The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). \includegraphics{figure_1} The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is 48 units\(^2\), determine the \(y\)-coordinate of \(P\). [7]

1. Find the first three terms in the expansion of \((1-2x)^{-1}\) in ascending powers of \(x\), where \(|x| < \frac{1}{2}\). [3]
2. Hence find the coefficient of \(x^2\) in the expansion of \(\frac{x+3}{\sqrt{1-2x}}\). [2]

Express \(\frac{9x^2+43x+8}{(3+x)(1-x)(2x+1)}\) in partial fractions. [4]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a constant.

1. Find \(\mathbf{A}^{-1}\). [2]

The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\).

1. Given that \(\mathbf{PA} = \mathbf{B}\), find the matrix \(\mathbf{P}\). [3]

1. \(P\), \(Q\) and \(T\) are three transformations in 2-D. \(P\) is a reflection in the \(x\)-axis. \(\mathbf{A}\) is the matrix that represents \(P\). Write down the matrix \(\mathbf{A}\). [1]
2. \(Q\) is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). \(\mathbf{B}\) is the matrix that represents \(Q\). Find the matrix \(\mathbf{B}\). [2]
3. \(T\) is \(P\) followed by \(Q\). \(\mathbf{C}\) is the matrix that represents \(T\). Determine the matrix \(\mathbf{C}\). [2]
4. \(L\) is the line whose equation is \(y = x\). Explain whether or not \(L\) is a line of invariant points under \(T\). [2]
5. An object parallelogram, \(M\), is transformed under \(T\) to an image parallelogram, \(N\). Explain what the value of the determinant of \(\mathbf{C}\) means about • the area of \(N\) compared to the area of \(M\). • the orientation of \(N\) compared to the orientation of \(M\). [3]

The equations of two lines are \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})\).

1. Show that these lines meet, and find the coordinates of the point of intersection. [5]
2. Find the acute angle between these lines. [3]

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 \(\text{Re}(z) \geq 9\). [6]

A locus \(C_1\) is defined by \(C_1 = \{z : |z + i| \leq |z - 2i|\}\).

1. Indicate by shading on the Argand diagram below the region representing \(C_1\). [2] \includegraphics{figure_8}
2. Find the cartesian equation of the boundary line of the region representing \(C_1\), giving your answer in the form \(ax + by + c = 0\). [2]

\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
2. Use the model to find the volume of water that the bottle can contain. [7]