Solve mixed sinh/cosh linear combinations

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

1. prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$$
2. find algebraically the exact solutions of the equation $$2 \sinh x + 7 \cosh x = 9$$ giving your answers as natural logarithms.

1. Solve the equation

$$18 \cosh x + 14 \sinh x = 11 + \mathrm { e } ^ { x }$$ Give your answers in the form \(\ln a\), where \(a\) is rational.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve the equation

$$7 \cosh x + 3 \sinh x = 2 \mathrm { e } ^ { x } + 7$$ Give your answers as simplified natural logarithms.

1. Find the eigenvalues of the matrix \(\left( \begin{array} { l l } 7 & 6 \\ 6 & 2 \end{array} \right)\)
2. Find the values of \(x\) for which

$$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms.
(Total 6 marks)

7

1. Using the definitions of hyperbolic functions in terms of exponentials, prove that $$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
2. Given that \(\cosh x \cosh y = 9\) and \(\sinh x \sinh y = 8\), show that \(x = y\).
3. Hence find the values of \(x\) and \(y\) which satisfy the equations given in part (ii), giving the answers in logarithmic form.

3 By first expressing \(\cosh x\) and \(\sinh x\) in terms of exponentials, solve the equation $$3 \cosh x - 4 \sinh x = 7$$ giving your answer in an exact logarithmic form.

11

1. Prove that \(\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } \equiv 2 \operatorname { coth } \theta\) Explicitly state any hyperbolic identities that you use within your proof.
   [0pt] [4 marks] LL
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2. Solve \(\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } = 4\) giving your answer in an exact form.
   [0pt] [2 marks]

5 The diagram shows part of the curve \(y = 5 \cosh x + 3 \sinh x\).
\includegraphics[max width=\textwidth, alt={}, center]{a6d9b3ec-5170-4f06-a8a3-b854efe36f07-3_496_771_315_246}

1. Solve the equation \(5 \cosh x + 3 \sinh x = 4\) giving your solution in exact form.
2. In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 1 } ( 5 \cosh x + 3 \sinh x ) \mathrm { d } x\) giving your answer in the form \(a \mathrm { e } + \frac { b } { \mathrm { e } }\) where \(a\) and \(b\) are integers to be determined.

9 Two thin poles, \(O A\) and \(B C\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \(( 0,3 ) , ( 5,0 )\) and \(( 2,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{c07ba83a-75fa-42dc-9bfd-6fc2f9226a23-5_805_1554_452_258} It is required to find the height of pole \(B C\) by modelling the shape of the curve that the chain forms.
Jofra models the curve using the equation \(\mathrm { y } = \mathrm { k } \cosh ( \mathrm { ax } - \mathrm { b } ) - 1\) where \(k , a\) and \(b\) are positive constants.

1. Determine the value of \(k\).
2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. Holly models the curve using the equation \(y = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 3\).
3. Write down the coordinates of the point, \(( u , v )\) where \(u\) and \(v\) are both non-zero, at which the two models will agree.
4. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(B C\) by 3.32 m to 3 significant figures.

1

1. Show that $$12 \cosh x - 4 \sinh x = 4 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x }$$
2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\).

1

1. Express $$5 \sinh x + \cosh x$$ in the form \(A \mathrm { e } ^ { x } + B \mathrm { e } ^ { - x }\), where \(A\) and \(B\) are integers.
2. Solve the equation $$5 \sinh x + \cosh x + 5 = 0$$ giving your answer in the form \(\ln a\), where \(a\) is a rational number.

1

1. Show that $$9 \sinh x - \cosh x = 4 \mathrm { e } ^ { x } - 5 \mathrm { e } ^ { - x }$$
2. Given that $$9 \sinh x - \cosh x = 8$$ find the exact value of \(\tanh x\).

1. (i) (a) Explain why \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is an improper integral.
   (b) Show that \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is divergent.
   (ii)

$$4 \sinh x = p \cosh x \quad \text { where } p \text { is a real constant }$$ Given that this equation has real solutions, determine the range of possible values for \(p\)

10. Find the values of \(x\) for which $$4 \cosh x + \sinh x = 8$$ giving your answer as natural logarithms.
[0pt] [P5 June 2003 Qn 1]

3 The curve \(C\) has equation $$y = \cosh x - 3 \sinh x$$

1. 1. The line \(y = - 1\) meets \(C\) at the point \(( k , - 1 )\). Show that $$\mathrm { e } ^ { 2 k } - \mathrm { e } ^ { k } - 2 = 0$$
   2. Hence find \(k\), giving your answer in the form \(\ln a\).
2. 1. Find the \(x\)-coordinate of the point where the curve \(C\) intersects the \(x\)-axis, giving your answer in the form \(p \ln a\).
   2. Show that \(C\) has no stationary points.
   3. Show that there is exactly one point on \(C\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

3 The diagram shows part of the curve \(y = 5 \cosh x + 3 \sinh x\).
\includegraphics[max width=\textwidth, alt={}, center]{ef967953-70b5-4dd1-a342-ad488b5fa79f-02_426_661_906_260}

1. Solve the equation \(5 \cosh x + 3 \sinh x = 4\) giving your solution in exact form.
2. In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 1 } ( 5 \cosh x + 3 \sinh x ) \mathrm { d } x\) giving your answer in the form \(a \mathrm { e } + \frac { b } { \mathrm { e } }\) where \(a\) and \(b\) are integers to be determined.