SPS SPS FM (SPS FM) 2022 February

1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
0 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 1 & 1
0 & - 1 \end{array} \right)\).

1. Find \(\mathbf { A } + 3 \mathbf { B }\).
2. Show that \(\mathbf { A } - \mathbf { B } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix and \(k\) is a constant whose value should be stated.

2. The complex numbers \(3 - 2 \mathrm { i }\) and \(2 + \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain these answers,

1. \(2 z - 3 w\),
2. \(( \mathrm { i } z ) ^ { 2 }\),
3. \(\frac { z } { w }\).

3. The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
\includegraphics[max width=\textwidth, alt={}, center]{bcba22d8-5f22-4576-b57c-7fdd05d128ad-1_344_349_993_1372}

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
2. Use integration to find the area of the shaded region bounded by the line and the curve.

4. The transformation S is a shear parallel to the \(x\)-axis in which the image of the point ( 1,1 ) is the point \(( 0,1 )\).

1. Draw a diagram showing the image of the unit square under S .
2. Write down the matrix that represents S .

5.

1. Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
2. Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
3. The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$ \section*{6.} In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.
4. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
5. Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).

7. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
0 & 1 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 2 }\) and \(\mathbf { A } ^ { 3 }\).
2. Hence suggest a suitable form for the matrix \(\mathbf { A } ^ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.

8.

1. Expand \(( 1 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }\) in ascending powers of \(x\).

9. The position vectors of three points \(A , B\) and \(C\) relative to an origin \(O\) are given respectively by $$\text { and } \quad \begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k } ,
& \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }
& \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } . \end{aligned}$$

1. Find the angle between \(A B\) and \(A C\).
2. Find the area of triangle \(A B C\).