4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials

241 questions

Sort by: Default | Easiest first | Hardest first
WJEC Further Unit 4 Specimen Q11
17 marks Challenging +1.3
  1. Show that $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad \text{where } -1 < x < 1.$$ [4]
  2. Given that $$a \cosh x + b \sinh x \equiv r \cosh(x + \alpha), \quad \text{where } a > b > 0,$$ show that $$\alpha = \frac{1}{2} \ln \left(\frac{a+b}{a-b}\right)$$ and find an expression for \(r\) in terms of \(a\) and \(b\). [7]
  3. Hence solve the equation $$5 \cosh x + 4 \sinh x = 10,$$ giving your answers correct to three significant figures. [6]
SPS SPS FM Pure 2021 May Q8
8 marks Challenging +1.3
  1. Using the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
  2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16u^3 + 12u = 3.$$ Give your answer in the form \(\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}\) where \(a\), \(b\) and \(c\) are integers. [5]
SPS SPS FM Pure 2022 February Q10
8 marks Standard +0.3
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]
SPS SPS FM 2021 November Q6
7 marks Challenging +1.8
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]
SPS SPS FM 2021 November Q7
7 marks Challenging +1.3
The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]
SPS SPS FM Pure 2023 February Q7
10 marks Challenging +1.3
  1. Prove that $$\tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad -k < x < k$$ stating the value of the constant \(k\). [5]
  2. Hence, or otherwise, solve the equation $$2x = \tanh\left(\ln \sqrt{2-3x}\right)$$ [5]
SPS SPS FM Pure 2024 February Q13
7 marks Challenging +1.2
In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh\frac{1}{2}x^2\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360°\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm. [7]
SPS SPS FM Pure 2026 November Q4
9 marks Challenging +1.2
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]
OCR Further Pure Core 1 2021 June Q4
9 marks Standard +0.3
  1. Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that $$x = \frac{1}{2}\ln\left(\frac{1+u}{1-u}\right).$$ [4]
  2. Solve the equation \(4\tanh^2 x + \tanh x - 3 = 0\), giving the solution in the form \(a\ln b\) where \(a\) and \(b\) are rational numbers to be determined. [4]
  3. Explain why the equation in part (b) has only one root. [1]
OCR Further Pure Core 2 2021 June Q5
11 marks Challenging +1.2
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]
OCR Further Pure Core 2 2018 March Q6
12 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Find the coordinates of all stationary points on the graph of \(y = 6\sinh^2 x - 13\cosh x\), giving your answers in an exact, simplified form. [9]
  2. By finding the second derivative, classify the stationary points found in part (i). [3]
OCR Further Pure Core 2 2018 December Q3
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the equation \(2\cosh^2 x + 5\sinh x - 5 = 0\) giving each answer in the form \(\ln(p + q\sqrt{r})\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. [6]
OCR Further Pure Core 2 2018 December Q9
5 marks Standard +0.8
  1. By using Euler's formula show that \(\cosh(\text{iz}) = \cos z\). [3]
  2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] [2]
Pre-U Pre-U 9795/1 2013 November Q13
24 marks Hard +2.3
  1. Let \(I_n = \int_0^{\alpha} \cosh^n x \, dx\) for integers \(n \geqslant 0\), where \(\alpha = \ln 2\).
    1. Prove that, for \(n \geqslant 2\), \(nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}\). [5]
    2. A curve has parametric equations \(x = 12 \sinh t + 4 \sinh^3 t\), \(y = 3 \cosh^4 t\), \(0 \leqslant t \leqslant \ln 2\). Find the length of the arc of this curve, giving your answer in the form \(a + b \ln 2\) for rational numbers \(a\) and \(b\). [8]
  2. The circle with equation \(x^2 + (y - R)^2 = r^2\), where \(r < R\), is rotated through one revolution about the \(x\)-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of \(\pi\), \(R\) and \(r\), the surface area of this torus. [11]
Pre-U Pre-U 9795/1 2018 June Q5
8 marks Standard +0.8
Find, in the form \(y = f(x)\), the solution of the differential equation \(\frac{dy}{dx} + y\tanh x = 2\cosh x\), given that \(y = \frac{3}{4}\) when \(x = \ln 2\). [8]
Pre-U Pre-U 9795 Specimen Q3
6 marks Standard +0.3
Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]