CAIE Further Paper 2 (Further Paper 2) 2020 November

1 Find the Maclaurin's series for \(\tan \left( x + \frac { 1 } { 4 } \pi \right)\) up to and including the term in \(x ^ { 2 }\).

2 A curve has equation \(\mathrm { y } = \cosh \mathrm { x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\).
Find, in terms of \(\pi\) and e, the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

3 Find all the roots of the equation \(( w + 1 ) ^ { 6 } = 1\), giving your answers in the form \(\mathrm { x } + \mathrm { iy }\) where \(x\) and \(y\) are real and exact.

4 Find the solution of the differential equation $$x \frac { d y } { d x } + 2 y = e ^ { x }$$ for which \(y = 3\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).

5 The curve \(C\) has equation $$y ^ { 2 } + ( x y + 1 ) ^ { 2 } = 5$$

1. Show that, at the point \(( 1,1 )\) on \(C , \frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 2 } { 3 }\).
2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 1,1 )\).

6 Find the particular solution of the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 8 \frac { d x } { d t } + 15 x = 102 \cos 3 t$$ given that, when \(t = 0 , x = 1\) and \(\frac { \mathrm { dx } } { \mathrm { dt } } = 0\).

7

1. Show that \(\sum _ { r = 1 } ^ { n } z ^ { 2 r } = \frac { z ^ { 2 n + 1 } - z } { z - z ^ { - 1 } }\), for \(z \neq 0,1 , - 1\).
2. By letting \(z = \cos \theta + i \sin \theta\), show that, if \(\sin \theta \neq 0\), $$1 + 2 \sum _ { r = 1 } ^ { n } \cos ( 2 r \theta ) = \frac { \sin ( 2 n + 1 ) \theta } { \sin \theta }$$

8
\includegraphics[max width=\textwidth, alt={}, center]{5b43cb39-7560-4484-ba6f-17303e986f47-10_369_1531_260_306} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } }\) for \(x \geqslant 0\), together with a set of \(n\) rectangles of unit width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + r + 1 } } < \ln \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } n + \frac { 2 } { 3 } \sqrt { n ^ { 2 } + n + 1 } \right)$$

9 It is given that \(a\) is a positive constant.

1. Show that the system of equations $$\begin{aligned} a x + ( 2 a + 5 ) y + ( a + 1 ) z & = 1
   - 4 y & = 2
   3 y - z & = 3 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & 2 a + 5 & a + 1
   0 & - 4 & 0
   0 & 3 & - 1 \end{array} \right)$$
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , - 1\) and - 4 .
3. Find a matrix \(\mathbf { P }\) such that $$\mathbf { A } = \mathbf { P } \left( \begin{array} { r r r } a & 0 & 0
   0 & - 1 & 0
   0 & 0 & - 4 \end{array} \right) \mathbf { P } ^ { - 1 } .$$
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.