SPS SPS FM (SPS FM) 2020 September

1. Vectors \(\overrightarrow { A B }\) and \(\overrightarrow { B C }\) are given by

$$\overrightarrow { A B } = \left( \begin{array} { r } 2 p
q
4 \end{array} \right) \quad \overrightarrow { B C } = \left( \begin{array} { r } q
- 3 p

2 \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\overrightarrow { A C }\) is parallel to \(\left( \begin{array} { r } 3
- 4
3 \end{array} \right)\), find the value of \(p\) and the value of \(q\).
2. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 3
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

4 \end{array} \right) \quad \overrightarrow { B C } = \left( \begin{array} { r } q
- 3 p
2 \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\overrightarrow { A C }\) is parallel to \(\left( \begin{array} { r } 3
- 4
3 \end{array} \right)\), find the value of \(p\) and the value of \(q\).
2. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 3
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)
   3. Using algebraic integration and making your method clear, find the exact value of $$\int _ { 1 } ^ { 5 } \frac { 4 x + 9 } { x + 3 } d x = a + \ln b$$ where \(a\) and \(b\) are constants to be found
   (4)
   4.
3. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a). Adele's working is shown below. \begin{displayquote} Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
   Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\) \end{displayquote} \begin{displayquote} Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\)
4. 1. Identify the mistake made by Adele in her working.
   2. Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\) \end{displayquote}

5. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 \cos 2 t , \quad y = 2 \tan t , \quad 0 \leq t \leq \frac { \pi } { 4 } .$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given $$\int _ { 0 } ^ { \frac { \pi } { 4 } } 24 \sin ^ { 2 } t \mathrm {~d} t$$
2. Hence, using algebraic integration, find the exact area of \(R\).

6. A sequence is defined by \(U _ { n } = 2 ^ { n + 1 } + 9 \times 13 ^ { n }\) for positive integer values of \(n\). Prove by induction that \(U _ { n }\) is divisible by 11 .
(5)

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f ^ { \prime } ( x ) = k - 4 x - 3 x ^ { 2 }$$ where \(k\) is constant,

1. show that \(C\) has a point of inflection at \(x = - \frac { 2 } { 3 }\) Given also that the distance \(A B = 4 \sqrt { 2 }\)
2. find, showing your working, the integer value of \(k\).
   (5)

9. Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$

10. A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis. Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

11. Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { c c } 2 & 0
4 & - 1 \end{array} \right)\)

12. Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h] \end{figure}

1. Find, in modulus-argument form, the complex number represented by the point P .
2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   • \(3 + 5 \mathrm { i }\)
   • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\)
     lie within this region.