CAIE Further Paper 2 (Further Paper 2) 2020 Specimen

1 Fid b g a ral sb t iord to d fferen ial eq tion $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 4 x = 72 \quad t ^ { 2 }$$

2 Fid \(\mathbf { b }\) ex ct le \(\mathbf { 6 } \int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\).

3 Fid b sb utiw the id fferen ial eq tin $$x \frac { \mathrm { dy } } { \mathrm { dx } } + 3 y = \frac { \sin x } { x }$$ fo wh ch \(y = O _ { N }\) b \(\mathrm { n } x = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).

4
\includegraphics[max width=\textwidth, alt={}, center]{6ff1b572-4cd8-433d-ba16-ffc8cda44476-06_545_958_264_552} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) fo \(x > 0\) tg th rwith a set \(6 ( n - 1 )\) rectab es 6 in t witd h

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < \frac { 2 n - 1 } { n }$$
2. Use a similar method to find, in terms of \(n\), a low er \(\mathbf { H }\)
   • \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }\).

5 Th cn e \(C\) has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { f } \mathbf { D } \quad 0 \leqslant t \leqslant 2$$

1. Find, in terms of e, the length of \(C\).
2. Find, in terms of \(\pi\) and \(e\), the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians ab the \(x\)-ax s.
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6

1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
2. Hen esh th the eq tion \(x ^ { 2 } - 4 x + 5 = 0\) s ro \(\operatorname { stan } ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) ad \(\operatorname { an } ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).

7

1. Starting from the definition of tanh in terms of exponentials, prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). [ \(\beta\)
2. Given that \(y = \operatorname { tah } ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right) , \mathrm { s } \quad\) th \(\mathrm { t } ( 2 x + 1 ) \frac { \mathrm { dy } } { \mathrm { dx } } + 1 = 0\)
3. Hence find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\) in the form $$a \ln 3 + b x + c x ^ { 2 }$$ wh re \(a , b\) ad \(c\) are constants to be determined.

8

1. 1. Fid bet basb le s a for which the system of equations $$\begin{array} { r l } x - 2 y - 2 z + z & 0
      2 x + ( a - 9 y - 0 z + 1 E & 0
      3 x - 6 y + 2 a z + 9 & 0 \end{array}$$ h san q sbtu in
   2. Given that \(a = - 3\), show that the system of equations in part (i) \(\mathbf { b } \mathbf { s } \mathbf { n }\) sb t in In erp et th s situation geometrically.
2. The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2
   0 & 2 & 2
   - 1 & 1 & 3 \end{array} \right)$$
   1. Find b eig le so A.
   2. Use th ch racteristic eq tiw \(\mathbf { A }\) tof id \(\mathbf { A } ^ { - 1 }\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.