Standard +0.8 This is a Further Maths hyperbolic equation requiring substitution using cosh²x = 1 + sinh²x to form a quadratic, solving it, then converting sinh x values back to x using the definition sinh x = (e^x - e^{-x})/2, which leads to a quadratic in e^x. While systematic, it requires multiple non-trivial algebraic manipulations and careful handling of the logarithmic form with surds—more demanding than standard C3 work but follows a clear method once the approach is identified.
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]
Use of In formula for \(\sinh^{-1}\) or \(\cosh^{-1}\): Do not allow eg \(\cosh^{-1}(5/4\) or 10) unless this is rejected (NB eg ln(3 - √10) is not real). If using \(\cosh^{-1}\) "rogue" solutions must be convincingly rejected. Most likely to see ln(3 + √10)
Must be in the correct form but allow \(\ln\left(\frac{1 + \sqrt{5}}{2}\right)\).
\(p = \frac{1}{2}, q = \frac{1}{2}, r = 5\)
A1
\(p = -3, q = 1, r = 10\)
A1
\(x = \ln(-3 + \sqrt{10})\)
A1
$\cosh^2 x - \sinh^2 x = 1$ | M1 | Use of identity to leave an equation in either just coshx or just sinhx
$2\sinh^2 x + 5\sinh x - 3 = 0$ | M1 | Reduction to 3 term quadratic in sinhx or coshx: $4\cosh^4 x - 4\cosh^2 x + 50 = 0$
$\sinh x = \frac{1}{2}$ or $-3$ | A1 |
$x = \ln\left(\frac{1}{2} + \sqrt{\frac{5}{4}}\right)$ | A1 | Use of In formula for $\sinh^{-1}$ or $\cosh^{-1}$: Do not allow eg $\cosh^{-1}(5/4$ or 10) unless this is rejected (NB eg ln(3 - √10) is not real). If using $\cosh^{-1}$ "rogue" solutions must be convincingly rejected. Most likely to see ln(3 + √10)
$x = \ln\left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)$ | A1 | Must be in the correct form but allow $\ln\left(\frac{1 + \sqrt{5}}{2}\right)$.
$p = \frac{1}{2}, q = \frac{1}{2}, r = 5$ | A1 |
$p = -3, q = 1, r = 10$ | A1 |
$x = \ln(-3 + \sqrt{10})$ | A1 |
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\textbf{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]
\hfill \mbox{\textit{OCR Further Pure Core 2 2018 Q3 [6]}}