CAIE Further Paper 2 (Further Paper 2) 2022 November

1 Find the Maclaurin's series for \(\ln \left( 1 + \mathrm { e } ^ { x } \right)\) up to and including the term in \(x ^ { 2 }\).

2

1. Show that the system of equations $$\begin{aligned} & x - y + 2 z = 4
   & x - y - 3 z = a
   & x - y + 7 z = 13 \end{aligned}$$ where \(a\) is a constant, does not have a unique solution.
2. Given that \(a = - 5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(a \neq - 5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.

3 The curve \(C\) has parametric equations $$\mathrm { x } = \mathrm { e } ^ { \mathrm { t } } - \frac { 1 } { 3 } \mathrm { t } ^ { 3 } , \quad \mathrm { y } = 4 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { t } } ( \mathrm { t } - 2 ) , \quad \text { for } 0 \leqslant t \leqslant 2$$ Find, in terms of e , the length of \(C\).

4

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1 .$$
2. Show that \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \tan ^ { - 1 } ( \sinh x ) \right) = \operatorname { sech } x\).
3. Sketch the graph of \(y = \operatorname { sechx }\), stating the equation of the asymptote.
4. By considering a suitable set of \(n\) rectangles of unit width, use your sketch to show that $$\sum _ { r = 1 } ^ { n } \operatorname { sechr } < \tan ^ { - 1 } ( \operatorname { sinhn } )$$
5. Hence state an upper bound, in terms of \(\pi\), for \(\sum _ { r = 1 } ^ { \infty }\) sech \(r\).

5 Find the particular solution of the differential equation $$2 \frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + y = 4 x ^ { 2 } + 3 x + 3$$ given that, when \(x = 0 , y = \frac { d y } { d x } = 0\).

6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 2 & - 3 & - 7
0 & 5 & 7
0 & 0 & - 2 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$ where \(a\) and \(b\) are integers to be determined.

7

1. State the sum of the series \(1 + \mathrm { w } + \mathrm { w } ^ { 2 } + \mathrm { w } ^ { 3 } + \ldots + \mathrm { w } ^ { \mathrm { n } - 1 }\), for \(w \neq 1\).
2. Show that \(( 1 + i \tan \theta ) ^ { k } = \sec ^ { k } \theta ( \cos k \theta + i \sin k \theta )\), where \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
3. By considering \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { \mathrm { k } }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \sec ^ { k } \theta \sin k \theta = \cot \theta \left( 1 - \sec ^ { n } \theta \cos n \theta \right)$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
4. Hence find \(\sum _ { k = 0 } ^ { 6 m - 1 } 2 ^ { k } \sin \left( \frac { 1 } { 3 } k \pi \right)\) in terms of \(m\).

8

1. Use the substitution \(u = 1 - ( \theta - 1 ) ^ { 2 }\) to find $$\int \frac { \theta - 1 } { \sqrt { 1 - ( \theta - 1 ) ^ { 2 } } } \mathrm {~d} \theta$$
2. Find the solution of the differential equation $$\theta \frac { d y } { d \theta } - y = \theta ^ { 2 } \sin ^ { - 1 } ( \theta - 1 ) ,$$ where \(0 < \theta < 2\), given that \(y = 1\) when \(\theta = 1\). Give your answer in the form \(y = \mathrm { f } ( \theta )\).
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