CAIE Further Paper 2 (Further Paper 2) 2022 June

1 Find the roots of the equation \(z ^ { 3 } = 7 \sqrt { 3 } - 7 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi \leqslant \theta < \pi\).

2

1. Find the coefficient of \(x ^ { 2 }\) in the Maclaurin's series for \(- \ln \cos x\).
2. Find the length of the arc of the curve with equation \(\mathrm { y } = - \operatorname { Incos } \mathrm { x }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 4 } \pi\).

3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 9 & 5
5 & - 8 & 5
1 & - 1 & 2 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }\), where \(p\) and \(q\) are constants to be determined.

4 It is given that $$x = - t + \tan ^ { - 1 } t \quad \text { and } \quad y = t + \sinh ^ { - 1 } t$$

1. Show that \(\frac { d y } { d x } = - \frac { t ^ { 2 } + 1 + \sqrt { t ^ { 2 } + 1 } } { t ^ { 2 } }\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) when \(t = \frac { 3 } { 4 }\).

5 Find the solution of the differential equation $$x ( x + 7 ) \frac { d y } { d x } + 7 y = x$$ for which \(y = 7\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).

6
\includegraphics[max width=\textwidth, alt={}, center]{23b06b1c-997f-425d-ae3d-bd4cc1295605-10_771_1146_260_497} The diagram shows the curve with equation \(\mathrm { y } = \ln ( 1 + \mathrm { x } )\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles each of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) d x < U _ { n }\), where $$U _ { n } = \frac { 1 } { n } \ln \frac { ( 2 n ) ! } { n ! } - \ln n$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(\mathrm { L } _ { \mathrm { n } }\) for \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) \mathrm { d } x\).
3. By simplifying \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } }\), show that \(\lim _ { \mathrm { n } \rightarrow \infty } \left( \mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).

7 The variables \(x\) and \(y\) are related by the differential equation $$4 \frac { d ^ { 2 } y } { d x ^ { 2 } } - y = 3$$ It is given that, when \(x = 0 , y = - 3\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2\).

1. Find \(y\) in terms of \(x\).
2. Deduce the exact value of \(x\) for which \(y = 0\). Give your answer in logarithmic form.

8

1. Find \(\int \sin \theta \cos ^ { n } \theta d \theta\), where \(n \neq - 1\).
   Let \(I _ { m , n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { m } \theta \cos ^ { n } \theta d \theta\).
2. Show that, for \(m \geqslant 2\) and \(n \geqslant 0\), $$I _ { m , n } = \frac { m - 1 } { m + n } I _ { m - 2 , n }$$
3. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that $$\cos ^ { 5 } \theta = a \cos 5 \theta + b \cos 3 \theta + c \cos \theta$$ where \(a\), \(b\) and \(c\) are constants to be determined.
4. Using the results given in parts (b) and (c), find the exact value of \(I _ { 2,5 }\).
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