CAIE Further Paper 2 (Further Paper 2) 2021 June

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1. Given that \(a\) is an integer, show that the system of equations $$\begin{aligned} a x + 3 y + z & = 14
   2 x + y + 3 z & = 0
   - x + 2 y - 5 z & = 17 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
2. Find the value of \(a\) for which \(x = 1 , y = 4 , z = - 2\) is the solution to the system of equations in part (a).

2 The variables \(x\) and \(y\) are related by the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 \frac { d y } { d x } + 2 y = 2 x + 1$$

1. Find the general solution for \(y\) in terms of \(x\).
2. State an approximate solution for large positive values of \(x\).

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\includegraphics[max width=\textwidth, alt={}, center]{fa2213b3-480c-44cb-8ba0-ebd2b94d3d90-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).

4 Find the solution of the differential equation $$\sin \theta \frac { d y } { d \theta } + y = \tan \frac { 1 } { 2 } \theta$$ where \(0 < \theta < \pi\), given that \(y = 1\) when \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( \theta )\). [You may use without proof the result that \(\int \operatorname { cosec } \theta d \theta = \ln \tan \frac { 1 } { 2 } \theta\).]

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1. State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\).
2. Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0\).
3. Given instead that \(z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$

6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 5 & - \frac { 22 } { 3 } & 8
0 & - 6 & 0
0 & 0 & 1 \end{array} \right)$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 3 }\).

7

1. It is given that \(\mathrm { y } = \operatorname { sech } ^ { - 1 } \left( \mathrm { x } + \frac { 1 } { 2 } \right)\).
   Express cosh \(y\) in terms of \(x\) and hence show that \(\sinh y \frac { d y } { d x } = - \frac { 1 } { \left( x + \frac { 1 } { 2 } \right) ^ { 2 } }\).
2. Find the first three terms in the Maclaurin's series for \(\operatorname { sech } ^ { - 1 } \left( x + \frac { 1 } { 2 } \right)\) in the form $$\ln a + b x + c x ^ { 2 }$$ where \(a\), \(b\) and \(c\) are constants to be determined.

8 The curve \(C\) has parametric equations $$\mathbf { x } = 2 \cosh t , \quad \mathbf { y } = \frac { 3 } { 2 } \mathbf { t } - \frac { 1 } { 4 } \sinh 2 \mathbf { t } , \text { for } 0 \leqslant t \leqslant 1$$

1. Find \(\frac { \mathrm { dx } } { \mathrm { dt } }\) and show that \(\frac { \mathrm { dy } } { \mathrm { dt } } = 1 - \sinh ^ { 2 } \mathrm { t }\).
   The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
2. 1. Show that \(\mathrm { A } = \pi \int _ { 0 } ^ { 1 } \left( \frac { 3 } { 2 } \mathrm { t } - \frac { 1 } { 4 } \sinh 2 \mathrm { t } \right) ( 1 + \cosh 2 \mathrm { t } ) \mathrm { dt }\).
   2. Hence find \(A\) in terms of \(\pi , \sinh 2\) and \(\cosh 2\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.