CAIE Further Paper 2 (Further Paper 2) 2023 November

1 Show that the system of equations $$\begin{aligned} 14 x - 4 y + 6 z & = 5
x + y + k z & = 3
- 21 x + 6 y - 9 z & = 14 \end{aligned}$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.

2 Find the roots of the equation \(( z + 5 i ) ^ { 3 } = 4 + 4 \sqrt { 3 } i\), giving your answers in the form \(r \cos \theta + i ( r \sin \theta - 5 )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

3 Find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).

4 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$ given that, when \(x = 0 , y = 2\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - 8\).

5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 2 } { 3 } \mathrm { t } ^ { \frac { 3 } { 2 } } - 2 \mathrm { t } ^ { \frac { 1 } { 2 } } , \quad \mathrm { y } = 2 \mathrm { t } + 5 , \quad \text { for } 0 < t \leqslant 3$$

1. Find the exact length of \(C\).
2. Find the set of values of \(t\) for which \(\frac { d ^ { 2 } y } { d x ^ { 2 } } > 0\).

6

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\sinh 2 x = 2 \sinh x \cosh x$$ \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_67_1550_374_347}
   \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_58_1569_475_328}
   \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_58_1569_566_328}
   \includegraphics[max width=\textwidth, alt={}]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_54_1566_657_328} ....................................................................................................................................................... .
   \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_54_1570_840_324}
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2. Using the substitution \(\mathrm { u } = \sinh \mathrm { x }\), find \(\int \sinh ^ { 2 } 2 x \cosh x \mathrm { dx }\).
3. Find the particular solution of the differential equation $$\frac { d y } { d x } + y \tanh x = \sinh ^ { 2 } 2 x$$ given that \(y = 4\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).

7 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } - 6 & 2 & 13
0 & - 2 & 5
0 & 0 & 8 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

8

1. State the sum of the series \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 }\), for \(z \neq 1\).
2. By letting \(z = \cos \theta + i \sin \theta\), where \(\cos \theta \neq 1\), show that $$1 + \cos \theta + \cos 2 \theta + \ldots + \cos ( n - 1 ) \theta = \frac { 1 } { 2 } \left( 1 - \cos n \theta + \frac { \sin n \theta \sin \theta } { 1 - \cos \theta } \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-15_833_785_214_680} The diagram shows the curve with equation \(\mathrm { y } = \cos \mathrm { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
3. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } \cos x d x < \frac { 1 } { 2 n } \left( 1 - \cos 1 + \frac { \sin 1 \sin \frac { 1 } { n } } { 1 - \cos \frac { 1 } { n } } \right)$$
4. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \cos x d x\).
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