CAIE Further Paper 2 (Further Paper 2) 2021 November

1 It is given that \(y = \sinh \left( x ^ { 2 } \right) + \cosh \left( x ^ { 2 } \right)\).

1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x ^ { 4 }\).
2. Deduce the value of \(\frac { \mathrm { d } ^ { 4 } \mathrm { y } } { \mathrm { dx } ^ { 4 } }\) when \(x = 0\).
3. Use your answer to part (a) to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { ydx }\), giving your answer as a rational
   fraction in its lowest terms. fraction in its lowest terms.

2 Find the solution of the differential equation $$\frac { d y } { d x } + \frac { 4 x ^ { 3 } y } { x ^ { 4 } + 5 } = 6 x$$ for which \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-04_867_812_278_621} The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 2 }\) 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 the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) d x < \frac { 4 n ^ { 2 } + 3 n - 1 } { 6 n ^ { 2 } }$$
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) \mathrm { dx }\).

4

1. Write down all the roots of the equation \(x ^ { 5 } - 1 = 0\).
2. Use de Moivre's theorem to show that \(\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1\).
3. Use the results of parts (a) and (b) to express each real root of the equation $$8 x ^ { 9 } - 8 x ^ { 7 } + x ^ { 5 } - 8 x ^ { 4 } + 8 x ^ { 2 } - 1 = 0$$ in the form \(\cos k \pi\), where \(k\) is a rational number.

5 The curve \(C\) has parametric equations $$x = 3 t + 2 t ^ { - 1 } + a t ^ { 3 } , \quad y = 4 t - \frac { 3 } { 2 } t ^ { - 1 } + b t ^ { 3 } , \quad \text { for } 1 \leqslant t \leqslant 2$$ where \(a\) and \(b\) are constants.

1. It is given that \(a = \frac { 2 } { 3 }\) and \(b = - \frac { 1 } { 2 }\). Show that \(\left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } = \frac { 25 } { 4 } \left( t ^ { 2 } + t ^ { - 2 } \right) ^ { 2 }\) and find the exact length of \(C\).
2. It is given instead that \(\mathrm { a } = \mathrm { b } = 0\). Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) when \(t = 1\).
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6 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 6 & 6
0 & 2 & 6
0 & 0 & - 3 \end{array} \right) .$$

1. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
2. Find the matrix \(\mathbf { A }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { l l l } 4 & 0 & 0
   0 & 5 & 0
   0 & 0 & 6 \end{array} \right) .$$
3. State the eigenvalues and corresponding eigenvectors of \(\mathbf { A } ^ { 3 }\).

7 It is given that \(y = x ^ { 2 } w\) and $$x ^ { 2 } \frac { d ^ { 2 } w } { d x ^ { 2 } } + 4 x ( x + 1 ) \frac { d w } { d x } + \left( 5 x ^ { 2 } + 8 x + 2 \right) w = 5 x ^ { 2 } + 4 x + 2$$

1. Show that $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 4 \frac { d y } { d x } + 5 y = 5 x ^ { 2 } + 4 x + 2$$
2. Find the general solution for \(w\) in terms of \(x\).

8

1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } x = \operatorname { sech } ^ { 2 } x$$
2. Using the substitution \(\mathrm { u } = \tanh \mathrm { x }\), or otherwise, find \(\int \operatorname { sech } ^ { 2 } x \tanh ^ { 2 } x \mathrm {~d} x\).
   It is given that, for \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \ln 3 } \operatorname { sech } ^ { \mathrm { n } } x \tanh ^ { 2 } x \mathrm { dx }\).
3. Show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = \left( \frac { 4 } { 5 } \right) ^ { 3 } \left( \frac { 3 } { 5 } \right) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$ [You may use the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \operatorname { sech } x ) = - \tanh x \operatorname { sech } x\).]
4. Find the value of \(I _ { 4 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.