SPS SPS FM (SPS FM) 2023 January

1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 2 & a
4 & 1 \end{array} \right)\). I denotes the \(2 \times 2\) identity matrix. Find

1. \(\mathbf { A } + 3 \mathbf { B } - 4 \mathbf { I }\),
2. \(\mathbf { A B }\).
   [0pt] [BLANK PAGE]

2. The transformations \(\mathrm { R } , \mathrm { S }\) and T are defined as follows.
R : reflection in the \(x\)-axis
S : stretch in the \(x\)-direction with scale factor 3
T: translation in the positive \(x\)-direction by 4 units

1. The curve \(y = \ln x\) is transformed by R followed by T . Find the equation of the resulting curve.
2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }\). You should make clear the order of the transformations.
   [0pt] [BLANK PAGE]

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

4. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines.

5.

1. Expand \(( 2 + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), and state the set of values of \(x\) for which the expansion is valid.
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).
   [0pt] [BLANK PAGE] \section*{6.} The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it.
   \includegraphics[max width=\textwidth, alt={}, center]{d193321f-0471-48cd-b954-4a7330777491-14_424_849_287_520} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
   (a) How many selections of seven cards are possible?
   (b) Find the probability that the seven cards include exactly one card showing the letter A .
   [0pt] [BLANK PAGE]

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = ( - 9 \mathbf { i } + 10 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
l _ { 2 } : & \mathbf { r } = ( 3 \mathbf { i } + \mathbf { j } + 17 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{array}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\) has position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\).
3. Show that \(A\) lies on \(l _ { 1 }\).
   [0pt] [BLANK PAGE]

8. $$\mathrm { f } ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\)

1. show all the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram,
2. find the value of \(p\) and the value of \(q\).
   [0pt] [BLANK PAGE]

9. Please remember to show detailed reasoning in your answer
\includegraphics[max width=\textwidth, alt={}, center]{d193321f-0471-48cd-b954-4a7330777491-20_467_817_239_639} The diagram shows the curve with equation \(y = ( 2 x - 3 ) ^ { 2 }\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis.
[0pt] [BLANK PAGE]

10. The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf { M } = \left( \begin{array} { c c } - 4 & - 4 \sqrt { 3 }
4 \sqrt { 3 } & - 4 \end{array} \right)$$

1. Determine
   1. the value of \(k\),
   2. the smallest value of \(\theta\) A square \(S\) has vertices at the points with coordinates \(( 0,0 ) , ( a , - a ) , ( 2 a , 0 )\) and \(( a , a )\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
2. Determine, in terms of \(a\), the area of \(S ^ { \prime }\)
   [0pt] [BLANK PAGE]

11. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set \(P\), of points that lie within the shaded region including its boundaries, is defined by $$P = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \}$$ where \(a , b , c\) and \(d\) are integers.

1. Write down the values of \(a , b , c\) and \(d\). The set \(Q\) is defined by $$Q = \{ z \in \mathbb { C } : a \leqslant | z + b + c \mathrm { i } | \leqslant d \} \cap \{ z \in \mathbb { C } : | z - \mathrm { i } | \leqslant | z - 3 \mathrm { i } | \}$$
2. Determine the exact area of the region defined by \(Q\), giving your answer in simplest form.
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]