Edexcel A-Level Mathematics A2 Fp3 1 2 Inverse Hyperbolic Functions Questions

Practise Edexcel IAL FP3.1.2 by solving the exponential quadratic for an inverse hyperbolic function and choosing its valid logarithmic branch.

Exam points

Derive inverse hyperbolic log forms by solving the exponential quadratic branch.
select the correct logarithmic branch from a stated sign or domain condition

[Maximum number: 6]

Given that

\cosh y=x \quad \text { and } \quad y<0

use the definition of $\cosh y$ in terms of exponential functions to prove that

y=\ln \left(x-\sqrt{x^{2}-1}\right)

$\cosh y=x,\quad y<0,\quad x=\frac{e^y+e^{-y}}{2}$ $2xe^y=e^{2y}+1 \Rightarrow e^{2y}-2xe^y+1=0$ $e^y=\frac{2x\pm\sqrt{(2x)^2-4}}{2} =x\pm\sqrt{x^2-1}$ $y=\ln\left(x-\sqrt{x^2-1}\right)$

Since y<0, $e^y<1$ , so the negative root is required.

B1 M1 M1 A1 A1* B1

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Edexcel A-Level Mathematics A2 Fp3 1 2 Inverse Hyperbolic Functions Questions question 1