CAIE Further Paper 2 (Further Paper 2) 2021 November

1 Find the Maclaurin's series for \(e ^ { x } \tan x\) from first principles up to and including the term in \(x ^ { 2 }\).

2 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 12
0 & 1 & 0
0 & 0 & 3 \end{array} \right) .$$ Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }$$ where \(p\) and \(q\) are integers to be determined.

3 The curve \(C\) has equation $$x y ^ { 3 } - 4 x ^ { 3 } y = 3$$

1. Show that, at the point \(( - 1,1 )\) on \(C , \frac { \mathrm { dy } } { \mathrm { dx } } = 11\).
2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( - 1,1 )\).
   \includegraphics[max width=\textwidth, alt={}, center]{37db1c60-0f94-413f-b29b-5872975eee9e-06_535_1584_276_276} The diagram shows the curve with equation \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } ^ { 2 } }\) for \(x \geqslant 2\), together with a set of \(( N - 2 )\) rectangles
   of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }\).

5 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + y = 4 \cos x$$ given that, when \(x = 0 , y = - 4\) and \(\frac { d y } { d x } = 3\).

6

1. Use de Moivre's theorem to show that $$\operatorname { cosec } 5 \theta = \frac { \operatorname { cosec } ^ { 5 } \theta } { 5 \operatorname { cosec } ^ { 4 } \theta - 20 \operatorname { cosec } ^ { 2 } \theta + 16 }$$
2. Hence obtain the roots of the equation $$x ^ { 5 } - 10 x ^ { 4 } + 40 x ^ { 2 } - 32 = 0$$ in the form \(\operatorname { cosec } ( q \pi )\), where \(q\) is rational.

7

1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$ is \(x + \sqrt { x ^ { 2 } - 1 }\).
2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$ for which \(y = 1\) when \(x = \frac { 5 } { 4 }\). Give your answer in the form \(y = f ( x )\).

8

1. Starting from the definition of cosh in terms of exponentials, prove that $$2 \cosh ^ { 2 } A = \cosh 2 A + 1$$ The curve \(C\) has parametric equations $$\mathrm { x } = 2 \cosh 2 \mathrm { t } + 3 \mathrm { t } , \quad \mathrm { y } = \frac { 3 } { 2 } \cosh 2 \mathrm { t } - 4 \mathrm { t } , \quad \text { for } - \frac { 1 } { 2 } \leqslant t \leqslant \frac { 1 } { 2 }$$ The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis is denoted by \(A\).
2. 1. Show that \(A = 10 \pi \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } ( 2 \cosh 2 t + 3 t ) \cosh 2 t d t\).
   2. Hence find \(A\) in terms of \(\pi\) and e.
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