Question 4 - A-Level Maths

Given that \(k \geqslant 1\) and \(\cosh x = k\), show that \(x = \pm \ln \left( k + \sqrt { k ^ { 2 } - 1 } \right)\).
Find \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\), giving the answer in an exact logarithmic form.
Solve the equation \(6 \sinh x - \sinh 2 x = 0\), giving the answers in an exact form, using logarithms where appropriate.
Show that there is no point on the curve \(y = 6 \sinh x - \sinh 2 x\) at which the gradient is 5 .