Intersection points of hyperbolic curves

5 The curves \(C _ { 1 } : y = \cosh x\) and \(C _ { 2 } : y = \sinh 2 x\) intersect at the point where \(x = a\).

1. Find the exact value of \(a\), giving your answer in logarithmic form.
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the exact value of the length of the arc of \(C _ { 1 }\) from \(x = 0\) to \(\mathrm { x } = \mathrm { a }\).

1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that

$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.

7. \begin{figure}[h] \end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.

1

1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).

4 Which of the following graphs intersects the graph of \(y = \sinh x\) at exactly one point? Circle your answer. \(y = \operatorname { cosech } x\) \(y = \cosh x\) \(y = \operatorname { coth } x\) \(y = \operatorname { sech } x\)

Find the number of solutions of the equation \(\tanh x = \cosh x\) Circle your answer. [1 mark] \(0 \quad 1 \quad 2 \quad 3\)

In this question you must show detailed reasoning.

1. The curves with equations $$y = \frac{3}{4}\sinh x \text{ and } y = \tanh x + \frac{1}{5}$$ intersect at just one point \(P\)
   1. Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation $$15e^{4x} - 48e^{3x} + 32e^x - 15 = 0$$ [3]
   2. Show that \(e^x = 3\) is a solution of this equation. [1]
   3. Hence state the exact coordinates of \(P\). [1]
2. Show that $$\int_{-4}^{0} \frac{e^x}{x^2} dx = e^{-\frac{1}{4}}$$ [4]

Two thin poles, \(OA\) and \(BC\), 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{figure_5} It is required to find the height of pole \(BC\) by modelling the shape of the curve that the chain forms. Jofra models the curve using the equation \(y = k \cosh(ax - b) - 1\) where \(k\), \(a\) and \(b\) are positive constants.

1. Determine the value of \(k\). [2]
2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. [5]

Holly models the curve using the equation \(y = \frac{1}{4}x^2 - 3x + 3\).

1. Write down the coordinates of the point, \((u, v)\) where \(u\) and \(v\) are both non-zero, at which the two models will agree. [1]
2. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(BC\) by \(3.32\)m to 3 significant figures. [3]