Inverse hyperbolic functionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The inverse hyperbolic functions have the logarithmic forms $\operatorname{arsinh}x=\ln\left(x+\sqrt{x^2+1}\right)$ for all real $x$, $\operatorname{arcosh}x=\ln\left(x+\sqrt{x^2-1}\right)$ for $x\ge1$, and $\operatorname{artanh}x=\frac12\ln\frac{1+x}{1-x}$ for $|x|<1$.

   (a)

   Find the exact value of $\operatorname{arsinh}\frac34$.

   [1 mark]

   • A$\ln\frac54$
   • B$\ln2$
   • C$-\ln2$
   • D$\ln\frac74$

   (b)

   Find the exact value of $\operatorname{artanh}\frac12$.

   [1 mark]

   • A$\ln3$
   • B$\frac12\ln\frac13$
   • C$\ln\frac32$
   • D$\frac12\ln3$

   (c)

   Find the exact value of $\operatorname{arcosh}5$, giving your answer in the form $\ln\left(a+b\sqrt6\right)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=\operatorname{arcosh}x$.

   (a)

   What is the largest possible domain of $f$?

   [1 mark]

   • A$x\ge1$
   • Ball real $x$
   • C$|x|<1$
   • D$x\ge0$

   (b)

   Find the value of $x$ for which $f(x)=\ln3$.

   [1 mark]

   • A$\frac43$
   • B$3$
   • C$\frac53$
   • D$\frac{10}{3}$

   (c)

   Explain how the graph of $y=\operatorname{arcosh}x$ is related to the graph of $y=\cosh x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For real $x$, $y=\operatorname{arsinh}x$ means that $x=\sinh y$.

   (a)

   Prove that $\operatorname{arsinh}x=\ln\left(x+\sqrt{x^2+1}\right)$.

   [3 marks]

   (b)

   Using the result in part (a), show that $\operatorname{arsinh}(-x)=-\operatorname{arsinh}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For $|x|<1$, $y=\operatorname{artanh}x$ means that $x=\tanh y$.

   (a)

   Prove that $\operatorname{artanh}x=\frac12\ln\frac{1+x}{1-x}$, and explain why the result needs $|x|<1$.

   [6 marks]

   (b)

   Using logarithmic forms, solve $\operatorname{artanh}x=\operatorname{arsinh}\frac43-\ln2$, giving the exact value of $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions