FP3 June 2015 Q1
1. Solve the equation \[2\cosh^2 x - 3\sinh x = 1\] giving your answers in terms of natural logarithms. (6)
| Scheme | Marks |
|---|---|
| \(2\left(1 + \sinh^2 x\right) - 3\sinh x = 1\) | M1 |
| \(2\sinh^2 x - 3\sinh x + 1 = 0\) | A1 |
| \((2\sinh x - 1)(\sinh x - 1) = 0\) | M1 |
| \(\sinh x\) or \(\dfrac{e^x - e^{-x}}{2} = \dfrac{1}{2}\) or 1 | A1 |
| \(x = \ln\dfrac{1}{2}\left(1 + \sqrt{5}\right),\ \ln\left(1 + \sqrt{2}\right)\) | A1, A1 M1A1 on ePEN |
| (6) | |
| (6 marks) |
Notes
M1: Attempt to use \(\cosh^2 x = 1 + \sinh^2 x\)
A1: Correct 3 term quadratic. The “= 0” may be implied by their attempt to solve.
M1: Attempts to solve their 3TQ = 0 leading to \(\sinh x = \ldots\) (= 0 may be implied)
A1: Both values correct
A1: \(x = \ln\dfrac{1}{2}\left(1 + \sqrt{5}\right)\) or \(\ln\left(1 + \sqrt{2}\right)\) oe
A1: \(x = \ln\dfrac{1}{2}\left(1 + \sqrt{5}\right)\) and \(\ln\left(1 + \sqrt{2}\right)\) oe and no other values
Allow equivalent answers e.g. \(\ln\left(\dfrac{1}{2} + \sqrt{\dfrac{5}{4}}\right),\ \ln\left(\dfrac{1}{2} + \sqrt{1 + \dfrac{1}{4}}\right)\) and allow awrt 3SF accuracy e.g. \(\ln 1.62,\ \ln 2.41\)
Alternative
| Scheme | Marks |
|---|---|
| \(2\left(\dfrac{e^x + e^{-x}}{2}\right)^2 - 3\left(\dfrac{e^x - e^{-x}}{2}\right) = 1\) | M1 |
| \(e^{4x} - 3e^{3x} + 3e^x + 1 = 0\) | A1 |
| \(\left(e^{2x} - e^x - 1\right)\left(e^{2x} - 2e^x - 1\right) = 0 \Rightarrow e^x = \ldots\) | M1 |
| \(e^x = \dfrac{1 + \sqrt{5}}{2},\ \dfrac{2 + \sqrt{8}}{2}\) | A1 |
| \(x = \ln\dfrac{1}{2}\left(1 + \sqrt{5}\right),\ \ln\left(1 + \sqrt{2}\right)\) | A1, A1 M1A1 on ePEN |
M1: Substitutes correct definitions for \(\sinh x\) and \(\cosh x\) in terms of exponentials
A1: Correct quartic in \(e^x\)
M1: Solves their quartic as far as \(e^x = \ldots\) For the correct quartic there must be a recognisable attempt to solve e.g. the product of two 3TQ’s in \(e^x\) or if answers only are given, they must be correct (1.62, 2.41, and possibly (-0.618, -0.414)). For an incorrect quartic there must be a recognisable attempt to solve a quartic with at least 4 terms.
A1: Correct values for \(e^x\). Allow \(e^x = \dfrac{1 \pm \sqrt{5}}{2},\ \dfrac{2 \pm \sqrt{8}}{2}\) but no incorrect values. Allow awrt 1.62, 2.41
A1: \(x = \ln\dfrac{1}{2}\left(1 + \sqrt{5}\right)\) or \(\ln\left(1 + \sqrt{2}\right)\) oe
A1: \(x = \ln\dfrac{1}{2}\left(1 + \sqrt{5}\right)\) and \(\ln\left(1 + \sqrt{2}\right)\) oe and no other values. allow awrt 3SF accuracy e.g. \(\ln 1.62,\ \ln 2.41\)