SPS SPS FM (SPS FM) 2023 February

1. Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 }
\frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).
Use A and B to disprove the proposition: "Matrix multiplication is commutative".

2. A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
[0pt] [BLANK PAGE]

3. Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
[0pt] [BLANK PAGE]

4. (a) You are given that the matrix \(\left( \begin{array} { c c } 2 & 1
- 1 & 0 \end{array} \right)\) represents a transformation \(T\).
You are given that the line with equation \(y = k x\) is invariant under T. Determine the value of \(k\).
(b) Determine whether the line with equation \(y = k x\) in part above is a line of invariant points under T.
[0pt] [BLANK PAGE]

5. (a) Expand \(\sqrt { 1 + 2 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
(b) Hence expand \(\frac { \sqrt { 1 + 2 x } } { 1 + 9 x ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
(c) Determine the range of values of \(x\) tor which the expansion in part (b) is valid.
[0pt] [BLANK PAGE]

6. (a) The members of a team stand in a random order in a straight line for a photograph. There are four men and six women. Find the probability that all the men are next to each other.
(b) Find the probability that no two men are next to one another.
[0pt] [BLANK PAGE]

7. Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) 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 \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
3. Determine the length \(Q R\).
   [0pt] [BLANK PAGE]

8. In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).

1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).
   [0pt] [BLANK PAGE]

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 \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
3. Determine the length \(Q R\).
   [0pt] [BLANK PAGE]
   8. In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).
4. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
5. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).
   [0pt] [BLANK PAGE]
   9. \(O\) 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]{7126440a-3ac4-4dc5-b39f-09d7d6b5c87b-18_663_1166_541_210} Find the volume of the funnel according to the model.
   [0pt] [BLANK PAGE]
   10. A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s
   t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears.
   [0pt] [BLANK PAGE]

11. Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geq 9\).
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]