CAIE Further Paper 2 (Further Paper 2) 2024 June

1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).

2 The curve \(C\) has parametric equations $$x = \cosh t , \quad y = \sinh t , \quad \text { for } 0 < t \leqslant \frac { 3 } { 5 }$$ The length of \(C\) is denoted by \(s\).

1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 5 } } \sqrt { \cosh 2 t } \mathrm {~d} t\).
   \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-04_2714_37_143_2008}
2. By finding the Maclaurin's series for \(\sqrt { \cosh 2 t }\) up to and including the term in \(t ^ { 2 }\) ,deduce an approximation to \(s\) .

3 The curve \(C\) has equation $$x ^ { 3 } + 2 x y + 8 y ^ { 3 } = - 12$$

1. Show that, at the point \(( - 2 , - 1 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - \frac { 1 } { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-06_2714_37_143_2008}
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( - 2 , - 1 )\).

4
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_408_1433_296_315} The diagram shows the curve with equation \(y = x ^ { - 2 }\) for \(2 \leqslant x \leqslant N\) together with a set of ( \(N - 2\) ) rectangles of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } } > \frac { 3 } { 2 } - \frac { 1 } { N } + \frac { 1 } { N ^ { 2 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_2718_35_141_2012}
2. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }\).
3. Deduce lower and upper bounds for \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\).

5

1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 338 \sin t$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-10_2715_35_143_2012}
2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx R \sin ( t - \phi ) ,$$ where the constants \(R\) and \(\phi\) are to be determined.

6

1. Show that \(\sum _ { r = 1 } ^ { n } z ^ { 4 r } = \frac { z ^ { 4 n + 2 } - z ^ { 2 } } { z ^ { 2 } - z ^ { - 2 } }\), for \(z ^ { 2 } \neq z ^ { - 2 }\).
2. By letting \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin 2 \theta \neq 0\), $$\sum _ { r = 1 } ^ { n } \sin ( 4 r \theta ) = \frac { \cos 2 \theta - \cos ( 4 n + 2 ) \theta } { 2 \sin 2 \theta }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-12_2718_35_143_2012}

7

1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \frac { x } { 2 } \sqrt { x ^ { 2 } - 9 } - \frac { 9 } { 2 } \cosh ^ { - 1 } \frac { x } { 3 } \right) = \sqrt { x ^ { 2 } - 9 }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_67_1579_413_324}
   \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_77_1581_497_322}
2. Find the solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = x ^ { 2 } \sqrt { x ^ { 2 } - 9 }$$ given that \(y = 1\) when \(x = 3\). Give your answer in the form \(y = \mathrm { f } ( x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_2716_35_143_2012}

8 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) do not intersect and are both perpendicular to \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The line \(l\) intersects \(\Pi _ { 1 }\) at the point \(( 1,6,0 )\) and intersects \(\Pi _ { 2 }\) at the point \(( 3 , - 6,0 )\).

1. Find Cartesian equations of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
2. Express the vector equation of \(l\) in the form \(\left( \begin{array} { l } x
   y
   z \end{array} \right) = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be determined, and hence show that for points on \(l , \frac { 1 } { 2 } x + \frac { 1 } { 12 } y = 1\) and \(z = 0\).
   \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-16_2715_40_144_2008}
3. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 3 \lambda ^ { 2 } + \frac { 7 } { 4 } \lambda = 0\) and hence find the eigenvalues of \(\mathbf { A }\). The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 3
   1 & 2 & 3
   \frac { 1 } { 2 } & \frac { 1 } { 12 } & 0 \end{array} \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-17_194_1711_484_212}
4. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\), where \(n\) is a positive integer. [6]
   \includegraphics[max width=\textwidth, alt={}]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_65_1581_335_322} ........................................................................................................................................
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   \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_2718_35_144_2012} If you use the following page to complete the answer to any question, the question number must be clearly shown.