CAIE Further Paper 2 (Further Paper 2) 2024 June

1 Find the roots of the equation \(z ^ { 3 } = - 108 \sqrt { 3 } + 108\) i, giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

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

3 It is given that $$\mathrm { x } = \sin ^ { - 1 } \mathrm { t } \quad \text { and } \quad \mathrm { y } = \mathrm { tcos } ^ { - 1 } \mathrm { t } , \quad \text { for } 0 \leqslant t < 1 .$$

1. Show that \(\frac { d y } { d x } = - t + \sqrt { 1 - t ^ { 2 } } \cos ^ { - 1 } t\).
2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) in terms of \(t\).

4 It is given that, for \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \ln 3 } \operatorname { sech } ^ { \mathrm { n } } \mathrm { xdx }\).

1. Show that, for \(n \geqslant 2\), $$( n - 1 ) \mathrm { I } _ { n } = \left( \frac { 3 } { 5 } \right) ^ { n - 2 } \left( \frac { 4 } { 5 } \right) + ( n - 2 ) \mathrm { I } _ { n - 2 }$$ [You may use the result that \(\frac { \mathrm { d } } { \mathrm { dx } } ( \operatorname { sech } x ) = - \tanh x \operatorname { sech } x\).]
2. Find the value of \(I _ { 4 }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{bca7281b-a6a9-4b4c-94e5-3da2a561ad86-08_663_1152_260_452} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x < U _ { n }\), where $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x\).
3. Show that \(\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).

6

1. Show that \(( \cosh x + \sinh x ) ^ { \frac { 1 } { 2 } } = \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
2. Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } + 3 y = 5 ( \cosh x + \sinh x ) ^ { \frac { 1 } { 2 } }$$ given that, when \(x = 0 , y = 1\) and \(\frac { d y } { d x } = \frac { 4 } { 3 }\).

7

1. Use the substitution \(\mathrm { u } = 1 + \mathrm { x } ^ { 2 }\) to find $$\int \frac { x } { \sqrt { 1 + x ^ { 2 } } } d x$$
2. Find the solution of the differential equation $$x \frac { d y } { d x } - y = x ^ { 2 } \sinh ^ { - 1 } x$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).

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1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { c l } 6 x + a y & = 3
   2 x - y & = 1
   x + 5 y + 4 z & = 2 \end{array}$$ has a unique solution.
2. Show that the system of equations in part (a) is consistent for all values of \(a\).
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 0 & 0
   2 & - 1 & 0
   1 & 5 & 4 \end{array} \right)$$
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to show that $$( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
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