CAIE Further Paper 2 (Further Paper 2) 2024 November

1 Find the value of \(\int _ { 6 } ^ { 7 } \frac { 1 } { \sqrt { ( x - 5 ) ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer in the form \(\ln ( a + \sqrt { b } )\), where \(a\) and \(b\) are integers to be determined.

2 The curve \(C\) has equation $$4 y ^ { 2 } + 4 \ln ( x y ) = 1 .$$

1. Show that, at the point \(\left( 2 , \frac { 1 } { 2 } \right)\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - \frac { 1 } { 6 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-04_2718_35_107_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-05_2725_35_99_20}
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( 2 , \frac { 1 } { 2 } \right)\).

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

1. Use de Moivre's theorem to show that $$\cot 6 \theta = \frac { \cot ^ { 6 } \theta - 15 \cot ^ { 4 } \theta + 15 \cot ^ { 2 } \theta - 1 } { 6 \cot ^ { 5 } \theta - 20 \cot ^ { 3 } \theta + 6 \cot \theta } .$$ \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-08_2718_35_107_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-09_2723_33_99_22}
2. Hence obtain the roots of the equation $$x ^ { 6 } - 6 x ^ { 5 } - 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } - 6 x - 1 = 0$$ in the form \(\cot ( q \pi )\), where \(q\) is a rational number.

5 Find the particular solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x ^ { 2 }$$ given that, when \(x = 0 , y = \frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
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\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-12_2720_38_109_2009}
(b) Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { 1 - x } \mathrm {~d} x\).
(c) Show that \(\lim _ { n \rightarrow \infty } \left( U _ { n } - L _ { n } \right) = 0\).
(d) Use the Maclaurin's series for \(\mathrm { e } ^ { x }\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z \left( 1 - \mathrm { e } ^ { - \frac { 1 } { z } } \right)\), in ascending powers of \(\frac { 1 } { z }\), and deduce the value of \(\lim _ { n \rightarrow \infty } \left( U _ { n } \right)\).

7

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \ln ( \tanh x ) ) = 2 \operatorname { cosech } 2 x\).
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-14_2717_35_106_2015}
   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-15_2723_33_99_22}
2. Find the solution of the differential equation $$\sinh 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = \sinh 2 x$$ for which \(y = 5\) when \(x = \ln 2\). Give your answer in an exact form.

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 0 & 0
0 & 7 & 9
4 & 1 & 7 \end{array} \right)$$

1. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 12 \lambda ^ { 2 } + 12 \lambda + 80 = 0\) and find the eigenvalues of A.
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   \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-17_2723_33_99_22}
2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { A } + r \mathbf { I } ,$$ where \(p , q\) and \(r\) are integers to be determined.
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 3 \mathbf { I } ) ^ { 4 } = \mathbf { P D P } ^ { - 1 }\) .
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