Questions - CAIE Further Paper 2

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

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

Starting from the definition of tanh in terms of exponentials, prove that $\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$. [ $\beta$
Given that $y = \operatorname { tah } ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right) , \mathrm { s } \quad$ th $\mathrm { t } ( 2 x + 1 ) \frac { \mathrm { dy } } { \mathrm { dx } } + 1 = 0$
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.

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.
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.

Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is $\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }$ .
\includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which $y = 6$ when $x = 3$.

By considering the binomial expansion of $\left( z + \frac { 1 } { z } \right) ^ { 7 }$ ,where $z = \cos \theta + \mathrm { i } \sin \theta$ ,use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where $a , b , c$ and $d$ are constants to be determined.
Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta$.
Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-18_2718_42_107_2007}
Using the results given in parts (a) and (b), find the exact value of $I _ { 9 }$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is $\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }$ .
\includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which $y = 6$ when $x = 3$.

By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
Use a similar method to find, in terms of $n$, a lower bound for $\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x$.

By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) d x \leqslant \frac { 3 n ^ { 2 } + 2 n - 1 } { 4 n ^ { 2 } }$$
Use a similar method to find, in terms of $n$, a lower bound for $\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) \mathrm { dx }$.

By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
Use a similar method to find, in terms of $N$, a lower bound for $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }$.

Questions — CAIE Further Paper 2 (186 questions)