CAIE Further Paper 2 (Further Paper 2) 2022 November

1

1. Find the set of values of \(k\) for which the system of equations $$\begin{aligned} x + 2 y + 3 z & = 1
   k x + 4 y + 6 z & = 0
   7 x + 8 y + 9 z & = 3 \end{aligned}$$ has a unique solution.
2. Interpret the situation geometrically in the case where the system of equations does not have a unique solution.

2 A curve has equation $$( x + 1 ) y + y ^ { 2 } = 2$$

1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 2 } { 3 }\) at the point \(( 0 , - 2 )\).
2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 0 , - 2 )\).

3

1. A curve has equation \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } + \frac { 1 } { 4 } \mathrm { e } ^ { - \mathrm { x } }\), for \(0 \leqslant x \leqslant 1\). 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.
2. Using standard results from the list of formulae (MF19), or otherwise, find the Maclaurin's series for \(\mathrm { e } ^ { x } + \frac { 1 } { 4 } \mathrm { e } ^ { - x }\) up to and including the term in \(x ^ { 2 }\).

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

5

1. Write down the fourth roots of unity.
2. Use de Moivre's theorem to show that $$\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1$$
3. Hence obtain the real roots of the equation $$16 \left( 8 x ^ { 4 } - 8 x ^ { 2 } + 1 \right) ^ { 4 } - 9 = 0$$ in the form \(\cos ( q \pi )\), where \(q\) is a rational number.

6
\includegraphics[max width=\textwidth, alt={}, center]{323ac7a5-4690-441d-87fc-325a393098fa-10_585_1349_258_358} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + 2 \mathrm { x } } }\) for \(x > 0\), together with a set of \(( n - 1 )\) rectangles of unit
width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + 2 r } } < \ln \left( n + 1 + \sqrt { n ^ { 2 } + 2 n } \right) + \frac { 1 } { 3 } \sqrt { 3 } - \ln ( 2 + \sqrt { 3 } )$$

7

1. It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector \(\mathbf { e }\). Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 3
   15 & - 4 & 3
   3 & 0 & 2 \end{array} \right)$$
2. Given that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
3. It is also given that \(\left( \begin{array} { l } 0
   1
   0 \end{array} \right)\) and \(\left( \begin{array} { l } 1
   2
   1 \end{array} \right)\) are eigenvectors of \(\mathbf { A }\). Find the corresponding eigenvalues.
4. Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
5. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q l\), where \(p\) and \(q\) are rational numbers to be determined.

8 It is given that \(\mathrm { y } = \operatorname { coshu }\), where \(u > 0\), and $$\sqrt { \cosh ^ { 2 } u - 1 } \left( \frac { d ^ { 2 } u } { d x ^ { 2 } } + \frac { d u } { d x } \right) + \cosh u \left( \frac { d u } { d x } \right) ^ { 2 } - 2 \cosh u = 4 e ^ { - x }$$

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