SPS SPS FM Pure (SPS FM Pure) 2024 January

Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(P(1, 2)\). \includegraphics{figure_1} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7]

1. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ where $$f(x) = k - |2x - 3k|$$ [2]

Find the value of the integral: $$\int_0^1 \frac{x^{\frac{1}{2}} + x^{-\frac{1}{3}}}{x} \, dx$$ [4]

The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.

1. Find the value of \(c\) and the value of \(d\). [3]

The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.

1. Find the position vector of \(P\). [6]
2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]

Let $$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$

1. Express \(f(x)\) in terms of partial fractions [5]
2. Hence, or otherwise, find the series expansion of \(f(x)\), in ascending powers of \(x\), up to and including the term in \(x^2\). Simplify each term. [6]
3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac{1}{2}\). [2]

$$\mathbf{M} = \begin{pmatrix} -2 & 5 \\ 6 & k \end{pmatrix}$$ where \(k\) is a constant. Given that $$\mathbf{M}^2 + 11\mathbf{M} = a\mathbf{I}$$ where \(a\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix,

1. 1. determine the value of \(a\)
   2. show that \(k = -9\) [3]
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf{M}\). [6]
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer. [1]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg(z - 6) = -\frac{3\pi}{4}$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies. [4]
2. Find the complex number for which both \(|z - 6i| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]