CAIE Further Paper 2 (Further Paper 2) 2020 June

1 Find the general solution of the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } - 8 \frac { d x } { d t } - 9 x = 9 e ^ { 8 t }$$

2 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } ( 1 + 3 \mathrm { x } ) ^ { \mathrm { n } } \mathrm { e } ^ { - 3 \mathrm { x } } \mathrm { dx }\), where \(n\) is an integer.

1. Show that \(3 \mathrm { I } _ { \mathrm { n } } = 1 - 4 ^ { \mathrm { n } } \mathrm { e } ^ { - 3 } + 3 \mathrm { nl } _ { \mathrm { n } - 1 }\).
2. Find the exact value of \(I _ { 2 }\).

3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 1 & 7
0 & 6 & 0
7 & 7 & 5 \end{array} \right) .$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-06_568_1614_294_262} The diagram shows the curve with equation \(\mathrm { y } = \ln \mathrm { x }\) for \(x \geqslant 1\), together with a set of ( \(N - 1\) ) rectangles of unit width.
3. By considering the sum of the areas of these rectangles, show that $$\ln N ! > N \ln N - N + 1 .$$
4. Use a similar method to find, in terms of \(N\), an upper bound for \(\operatorname { In } N\) !.

5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 1 } { 2 } \mathrm { t } ^ { 2 } - \ln \mathrm { t } , \quad \mathrm { y } = 2 \mathrm { t } + 1 , \quad \text { for } \frac { 1 } { 2 } \leqslant t \leqslant 2$$

1. Find the exact length of \(C\).
2. Find \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) in terms of \(t\), simplifying your answer.

6

1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } \theta = \operatorname { sech } ^ { 2 } \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1552_374_347}
   \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_67_1569_466_328} ......................................................................................................................................... ........................................................................................................................................
   \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_735_324}
   \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_826_324} .......................................................................................................................................... . ........................................................................................................................................ . The variables \(x\) and \(y\) are such that \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < x < \frac { 3 } { 4 } \pi\).
2. By differentiating the equation \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\) with respect to \(x\), show that $$\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosec } \left( \mathrm { x } + \frac { 1 } { 4 } \pi \right)$$
3. Hence find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \cos \left( x + \frac { 1 } { 4 } \pi \right) \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).

7

1. Show that an appropriate integrating factor for $$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$ is \(x + \sqrt { x ^ { 2 } + 1 }\).
2. Hence find the solution of the differential equation $$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$ for which \(y = \ln 2\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).

8

1. Use de Moivre's theorem to show that \(\sin ^ { 6 } \theta = - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )\).
   It is given that \(\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 6 } \left( \frac { 1 } { 4 } x \right) + \sin ^ { 6 } \left( \frac { 1 } { 4 } x \right) \right) \mathrm { d } x\).
3. Express each root of the equation \(16 c ^ { 6 } + 16 \left( 1 - c ^ { 2 } \right) ^ { 3 } - 13 = 0\) in the form \(\cos k \pi\), where \(k\) is a rational number.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.