CAIE Further Paper 2 (Further Paper 2) 2023 November

1 Find the Maclaurin's series for \(\ln ( x + 2 ) + \ln \left( x ^ { 2 } + 5 \right)\) up to and including the term in \(x ^ { 2 }\).

2 It is given that $$x = 1 + \frac { 1 } { t } \quad \text { and } \quad y = t e ^ { t }$$

1. Show that \(\frac { d y } { d x } = - e ^ { t } \left( t ^ { 3 } + t ^ { 2 } \right)\).
2. Find \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) in terms of \(t\).

3

1. Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$
2. Hence obtain the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x - \sqrt { 2 } = 0$$ in the form \(\cos ( q \pi )\), where \(q\) is a rational number.

4 Find the solution of the differential equation $$\frac { d y } { d x } + 3 y = \sin x$$ for which \(y = 1\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).

5
\includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-08_773_1161_278_443} The diagram shows part of the curve \(\mathrm { y } = \mathrm { xsech } ^ { 2 } \mathrm { x }\) and its maximum point \(M\).

1. Show that, at \(M\), $$2 x \tanh x - 1 = 0$$ and verify that this equation has a root between 0.7 and 0.8 .
2. By considering a suitable set of rectangles, use the diagram to show that
   \(\sum _ { r = 2 } ^ { n } r \operatorname { sech } ^ { 2 } r < n \tanh n + \operatorname { lnsechn } - \tanh 1 - \operatorname { lnsech } 1\).

6 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & - 1 & 1
0 & 2 & 1
0 & 0 & - 1 \end{array} \right) .$$

1. State the eigenvalues of \(\mathbf { P }\).
2. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
   The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct non-zero eigenvalues \(a , \frac { 1 } { 2 } , 2\) with corresponding eigenvectors $$\left( \begin{array} { l } 1
   0
   0 \end{array} \right) , \quad \left( \begin{array} { r } - 1
   2
   0 \end{array} \right) , \quad \left( \begin{array} { r } 1
   1
   - 1 \end{array} \right) ,$$ respectively.
3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

7

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$2 \sinh ^ { 2 } A = \cosh 2 A - 1$$ \includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_79_1556_358_347}
   \includegraphics[max width=\textwidth, alt={}]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_69_1575_466_328} ....................................................................................................................................... ........................................................................................................................................
2. A curve has equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
   Use the substitution \(\mathrm { X } = \frac { 1 } { 2 } \operatorname { sinhu }\) to show that \(S = \frac { 1 } { 32 } \pi \left( \frac { 820 } { 81 } - \ln 3 \right)\).

8 It is given that \(\mathbf { v } = y ^ { 4 }\) and $$y ^ { 3 } \frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 y ^ { 2 } \left( \frac { d y } { d x } \right) ^ { 2 } + y ^ { 3 } \frac { d y } { d x } + y ^ { 4 } = e ^ { - 2 x }$$

1. Show that $$\frac { d ^ { 2 } v } { d x ^ { 2 } } + \frac { d v } { d x } + 4 v = 4 e ^ { - 2 x }$$
2. Find \(y\) in terms of \(x\), given that, when \(x = 0 , y = 1\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 3 } { 8 }\).
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