SPS SPS FM Pure (SPS FM Pure) 2025 January

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

1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).
   (Total 8 marks)
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3. $$\mathbf { A } = \left( \begin{array} { c r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is constant. }$$ A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).

1. Find the value of \(k\) for which the line \(y = 2 x\) is mapped onto itself under \(T\).
2. Show that \(\mathbf { A }\) is non-singular for all values of \(k\).
3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
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4. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0
0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1
1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$

1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
2. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3
   3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
3. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).
   (3)
   [0pt] [BLANK PAGE] With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
   & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
5. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
6. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. 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 \(l _ { 2 }\)
7. Find the position vector of \(B\).
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6. You are given the complex number \(w = 2 + 2 \sqrt { 3 } i\).

1. Express \(w\) in modulus-argument form.
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).
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   (Total 12 marks)
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7. 7 A candlestick has base diamater 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. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} a Show that the volume generated by rotating the shaded region (in Figure 10) \(2 \pi\) radians about the \(y\)-axis is \(\frac { 112 } { 15 } \pi\)
b Hence find the volume of metal needed to create the candlestick.
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