Prove hyperbolic identity from exponentials

2

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh 2 x = 2 \sinh ^ { 2 } x + 1$$
2. Find the set of values of \(k\) for which \(\cosh 2 \mathrm { x } = \mathrm { ksinh } \mathrm { x }\) has two distinct real roots.

8

1. Starting from the definitions of sech and tanh in terms of exponentials, prove that $$1 - \operatorname { sech } ^ { 2 } t = \tanh ^ { 2 } t$$ \includegraphics[max width=\textwidth, alt={}]{d3ddf5ce-4399-4438-ab67-7bdb2e1bea6e-14_77_1547_360_347} ......................................................................................................................................... ......................................................................................................................................... . ........................................................................................................................................ ........................................................................................................................................ ....................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{d3ddf5ce-4399-4438-ab67-7bdb2e1bea6e-14_72_1573_911_324} \includegraphics[max width=\textwidth, alt={}, center]{d3ddf5ce-4399-4438-ab67-7bdb2e1bea6e-14_67_1570_1005_324} The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 1 } { 2 } \tanh ^ { 2 } \mathrm { t } + \text { Insecht } , \quad \mathrm { y } = 1 + \tanh ^ { 4 } \mathrm { t } , \quad \text { for } t > 0$$
2. Show that \(\frac { d y } { d x } = - 4 \operatorname { sech } ^ { 2 } t\).
3. Find the coordinates of the point on \(C\) with \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 9 } { 2 }\), giving your answer in the form \(( a + \ln b , c )\) where \(a , b\) and \(c\) are rational numbers.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8

1. Starting from the definitions of sech and tanh in terms of exponentials, prove that $$1 - \operatorname { sech } ^ { 2 } t = \tanh ^ { 2 } t$$ \includegraphics[max width=\textwidth, alt={}]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-14_77_1547_360_347} ......................................................................................................................................... ......................................................................................................................................... . ........................................................................................................................................ ........................................................................................................................................ ....................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-14_72_1573_911_324} \includegraphics[max width=\textwidth, alt={}, center]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-14_67_1573_1005_324} The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 1 } { 2 } \tanh ^ { 2 } \mathrm { t } + \text { Insecht } , \quad \mathrm { y } = 1 + \tanh ^ { 4 } \mathrm { t } , \quad \text { for } t > 0$$
2. Show that \(\frac { d y } { d x } = - 4 \operatorname { sech } ^ { 2 } t\).
3. Find the coordinates of the point on \(C\) with \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 9 } { 2 }\), giving your answer in the form \(( a + \ln b , c )\) where \(a , b\) and \(c\) are rational numbers.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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} \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_53_1570_932_324} \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_53_1570_1023_324}
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 \(y = f ( x )\).

1

1. Starting from the definition of \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
2. Given that \(\cosh 2 x = k\), where \(k > 1\), express each of \(\cosh x\) and \(\sinh x\) in terms of \(k\).

1. (a) Using the definition of \(\sinh x\) in terms of exponentials, prove that

$$4 \sinh ^ { 3 } x + 3 \sinh x \equiv \sinh 3 x$$ (b) Hence solve the equation $$\sinh 3 x = 19 \sinh x$$ giving your answers as simplified natural logarithms where appropriate.

6 Prove the identity $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$

1 Express \(\cosh ^ { 2 } x\) in terms of \(\sinh x\) Circle your answer. \(1 + \sinh ^ { 2 } x\) \(1 - \sinh ^ { 2 } x\) \(\sinh ^ { 2 } x - 1\) \(- 1 - \sinh ^ { 2 } x\)

14

1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}

1. The continuous random variable \(X\) has probability density function given by

$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the probability density function \(\mathrm { f } ( x )\) The graph consists of two straight line segments of equal length joined at the point where \(x = 4\)

1. Show that \(a = \frac { 1 } { 16 }\)
2. Hence find
   1. the value of \(b\)
   2. the value of \(c\)
3. Using algebraic integration, show that \(\operatorname { Var } ( X ) = \frac { 8 } { 3 }\)
4. Find, to 2 decimal places, the lower quartile and the upper quartile of \(X\) A statistician claims that $$\mathrm { P } ( - \sigma < X - \mu < \sigma ) > 0.5$$ where \(\mu\) and \(\sigma\) are the mean and standard deviation of \(X\)
5. Show that the statistician's claim is correct.

Prove that \(\sinh(i\pi - \theta) = \sinh \theta\). [4]

Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of exponentials,

1. prove that \(\cosh^2 x - \sinh^2 x = 1\), [3]
2. solve \(\operatorname{cosech} x - 2 \coth x = 2\), giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are integers. [4]

Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that \(\cosh^2 x - \sinh^2 x = 1\) [2 marks]