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.

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.

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. 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 \(p\)

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.

4

1. Using the definitions of sinh and cosh in terms of exponentials, prove that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
2. Solve the equation \(5 \cosh x + 3 \sinh x = 12\), giving your answers in the form \(\ln ( p \pm q \sqrt { 2 } )\) for rational numbers \(p\) and \(q\) to be determined.

3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.

The curve \(C_1\) has equation \(y = 3\sinh 2x\), and the curve \(C_2\) has equation \(y = 13 - 3e^{2x}\).

1. Sketch the graph of the curves \(C_1\) and \(C_2\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. [4]
2. Solve the equation \(3\sinh 2x = 13 - 3e^{2x}\), giving your answer in the form \(\frac{1}{2}\ln k\), where \(k\) is an integer. [5]

Using the definitions of hyperbolic functions in terms of exponentials,

1. show that $$\operatorname{sech}^2 x = 1 - \tanh^2 x$$ [3]
2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$ [4]

Solve the equation $$10 \cosh x + 2 \sinh x = 11.$$ Give each answer in the form \(\ln a\) where \(a\) is a rational number. [7]

Find the values of \(x\) for which $$4 \cosh x + \sinh x = 8,$$ giving your answer as natural logarithms. [6]

Find the values of \(x\) for which $$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms. (Total 6 marks)

1. Solve the equation $$\sinh t + 7 \cosh t = 8,$$ expressing your answer in exact logarithmic form. [6]

A curve has equation \(y = \cosh 2x + 7 \sinh 2x\).

1. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16. Show that there is no point on the curve at which the gradient is zero. Sketch the curve. [8]
2. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac{1}{2}\). [4]

You are given that \(\mathrm{f}(x) = 4 \sinh x + 3 \cosh x\).

1. Show that the curve \(y = \mathrm{f}(x)\) has no turning points. [3]
2. Determine the exact solution of the equation \(\mathrm{f}(x) = 5\). [5]

You are given that \(f(x) = 4\sinh x + 3\cosh x\).

1. Show that the curve \(y = f(x)\) has no turning points. [3]
2. Determine the exact solution of the equation \(f(x) = 5\). [5]

Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]