SPS SPS FM Pure (SPS FM Pure) 2024 January

1. Fig. 6 shows the region enclosed by part of the curve \(y = 2 x ^ { 2 }\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(\mathrm { P } ( 1,2 )\).

\begin{figure}[h] \end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed.
[0pt] [BLANK PAGE] \section*{2. a)} Find, in terms of \(k\), the set of values of \(x\) for which $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
b) Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$$ where $$\mathrm { f } ( x ) = k - | 2 x - 3 k |$$ [BLANK PAGE]

3. Find the value of the integral:
\(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).
(4 marks)
(Total 4 marks)
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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 \(\overrightarrow { O P }\) is perpendicular to \(l\), where \(O\) is the origin.
2. Find the position vector of \(P\).
   (6)
3. Find the area of triangle \(O A B\), giving your answer to 3 significant figures.
   (4)
   (Total 13 marks)
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5. Let $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) }$$

1. Express \(f ( x )\) in terms of partial fractions
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). Simplify each term.
3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac { 1 } { 2 }\).
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6. $$\mathbf { M } = \left( \begin{array} { r r } - 2 & 5
6 & k \end{array} \right)$$ 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 \(k = - 9\)
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf { M }\).
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer.
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7.

• The point \(P\) represents a complex number \(z\) on an Argand diagram such that

$$| z - 6 \mathrm { i } | = 2 | z - 3 |$$

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