Further Mathematics 9231: Hyperbolic functions Topical Past Paper Questions

The curve C has parametric equations $$ x=2\cosh2t+3t,\quad y=\frac{3}{2}\cosh2t-4t,\quad for0\leqslant t\leqslant1. $$ (a) Find, in terms of e, the length of C. [5] (b) Show that $$ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{a\cosh2t}{\left(b\sinh2t+c\right)^{n}}, $$ where $a…

(a) Find the exact values of x for which $ \cosh 2x = 6\sinh^{2}x $, giving your answers in logarithmic form. [4] (b) Sketch the curves $ C_{1}: y = \cosh 2x $ and $ C_{2}: y = 6\sinh^{2}x $ on the same diagram. [2]

A curve has equation $ y = e^{x} $ for $ \ln\frac{4}{3} \leqslant x \leqslant \ln\frac{12}{5} $. The area of the surface generated when the curve is rotated through $ 2\pi $ radians about the x-axis is denoted by A. (a) Use the substitution $u = e^{x}$ to show that $$ A=2\pi\int_…