SPS SPS FM (SPS FM) 2024 February

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 \(B C\) is part of the curve with equation \(x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37\).
The closed shape \(O A B C D\) is formed, in turn, from the line segments \(O A\) and \(A B\), the arc \(B C\) and the line segments \(C D\) and \(D O\) (see diagram).
A funnel can be modelled by rotating \(O A B C D\) by \(2 \pi\) radians about the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-04_510_894_443_226} Find the volume of the funnel according to the model.
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2. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\).

1. \includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-06_318_511_187_904} Give the coordinates of the points of intersection of the graphs with the coordinate axes.
2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).
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3. Show that \(\int _ { 4 } ^ { \infty } x ^ { - \frac { 3 } { 2 } } d x = 1\) [0pt]

4. Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11 \\ 10 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ - 2 \\ 1 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 2 \\ 4 \end{array} \right) \end{aligned}$$

+ \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) $$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(/ _ { 1 }\) and \(/ _ { 2 }\). \(Q\) is a point on \(/ 1\) which is 12 metres away from \(P \cdot R\) is the point on \(/ 2\) such that \(Q R\) is perpendicular to \(/ 1\).
3. Determine the length \(Q R\).
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5.

1. Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
2. 
   Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
3. State the set of values for which the expansion in part (b) is valid.
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6. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(= \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).

1. Find the value of a such that \(\mathbf { A B } = \mathbf { B A }\).
2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
3. A triangle of area 4 square units is transformed by the matrix \(\mathbf { B }\). Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
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$$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 10 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ - 2 \\ 1 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 2 \\ 4 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). \(Q\) is a point on \(l1\) which is 12 metres away from \(P \cdot R\) is the point on \(l2\) such that \(QR\) is perpendicular to \(l1\).
3. Determine the length \(QR\).
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   5. (a) Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
   (b)
   Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
   (c) State the set of values for which the expansion in part (b) is valid.
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   6. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).
   (a) Find the value of a such that \(\mathbf { A B } = \mathbf { B A }\).
   (b) Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
   (c) A triangle of area 4 square units is transformed by the matrix \(\mathbf { B }\). Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
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   7. (a) In this question you must show detailed reasoning. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
   (b) The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z | = | z - 2 i |\) and \(| z - 2 | = \sqrt { 5 }\) respectively.
   i. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\), showing any intercepts with the imaginary axis.
   ii. Indicate, by shading on your Argand diagram, the region \(\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}\).
   (c) i. Show that both of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfy \(| z - 2 | < \sqrt { 5 }\).
   ii. State, with a reason, which root of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfies \(| z | < | z - 2 i |\).
   (d) On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
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