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 \(y = f ( x )\).

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]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_67_1550_374_347}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_58_1569_475_328}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_58_1569_566_328}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_59_1566_657_328}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_58_1570_749_324}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_54_1570_840_324}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_63_1570_922_324}
   \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_67_1570_1009_324}
2. Using the substitution \(\mathrm { u } = \sinh \mathrm { x }\), find \(\int \sinh ^ { 2 } 2 x \cosh x \mathrm {~d} x\).
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 \(k\).

5
- 1
3 \end{array} \right] \quad \left[ \begin{array} { l } 2
1
1 \end{array} \right]$$ 2 Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\)\\[0pt] [2 marks]\\ 3

1. Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers \(A\) and \(B\)\\ 3
2. Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$ 4 A student states that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos x + \sin x } { \cos x - \sin x } \mathrm {~d} x\) is not an improper integral because \(\frac { \cos x + \sin x } { \cos x - \sin x }\) is defined at both \(x = 0\) and \(x = \frac { \pi } { 2 }\) Assess the validity of the student's argument.
   [0pt] [2 marks]
   \(5 \quad \mathrm { p } ( z ) = z ^ { 4 } + 3 z ^ { 2 } + a z + b , a \in \mathbb { R } , b \in \mathbb { R }\)
   \(2 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { p } ( \mathrm { z } ) = 0\) 5
3. Express \(\mathrm { p } ( z )\) as a product of quadratic factors with real coefficients.
   5
4. Solve the equation \(\mathrm { p } ( z ) = 0\).

2

1. Use the definitions of \(\cosh \theta\) and \(\sinh \theta\) in terms of \(\mathrm { e } ^ { \theta }\) to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
2. It is given that \(x\) satisfies the equation $$\cosh ( x - \ln 2 ) = \sinh x$$
   1. Show that \(\tanh x = \frac { 5 } { 7 }\).
   2. Express \(x\) in the form \(\frac { 1 } { 2 } \ln a\).

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.

7. Prove that \(\sinh ( \mathrm { i } \pi - \theta ) = \sinh \theta\).
[0pt] [P6 June 2002 Qn 1]

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}

12

1. Starting from the identities for \(\sinh 2 x\) and \(\cosh 2 x\), prove the identity $$\tanh 2 x = \frac { 2 \tanh x } { 1 + \tanh ^ { 2 } x }$$ 12
2. 1. The function f is defined by $$\mathrm { f } ( x ) = \tanh x \quad ( x > 0 )$$ State the range of f
      12
3. (ii) Use part (a) and part (b)(i) to prove that \(\tanh 2 x > \tanh x\) if \(x > 0\)