Inverse hyperbolic functionsAQA A-Level Further Maths: Subtopic test

1. 1
   Let $a=\sinh^{-1}\left(\frac34\right)$ and $b=\tanh^{-1}\left(\frac12\right)$.

   (a)

   Find the exact value of $a$.

   [1 mark]

   • A$\ln\frac{37}{16}$
   • B$\ln\frac74$
   • C$\ln\frac12$
   • D$\ln2$

   (b)

   Find the exact value of $b$.

   [1 mark]

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

   (c)

   Express $a+2b$ as a single natural logarithm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $c=\cosh^{-1}\left(\frac53\right)$.

   (a)

   Find the exact value of $c$.

   [1 mark]

   • A$\ln3$
   • B$\ln\frac{5+\sqrt{34}}{3}$
   • C$\ln\frac13$
   • D$\ln\frac{31}{9}$

   (b)

   Find the exact value of $\sinh c$.

   [1 mark]

   • A$\frac53$
   • B$\frac83$
   • C$\frac43$
   • D$\frac34$

   (c)

   Solve $\cosh x=\frac53$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student sets $y=\sinh^{-1}x$ for a real number $x$, so that $x=\sinh y$.

   (a)

   Show that $y=\ln\left(x+\sqrt{x^2+1}\right)$.

   [3 marks]

   (b)

   Hence show that $\sinh^{-1}(-x)=-\sinh^{-1}x$, and find the exact value of $\sinh^{-1}\left(-\frac34\right)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The inverse hyperbolic functions can be written using natural logarithms.

   (a)

   Let $y=\tanh^{-1}x$ for $-1<x<1$. Show that $y=\frac12\ln\left(\frac{1+x}{1-x}\right)$, and explain why the condition $-1<x<1$ is needed.

   [6 marks]

   (b)

   Find exact values, in terms of natural logarithms, for:
   (i) $\cosh^{-1}2$;
   (ii) $\tanh^{-1}\left(-\frac12\right)$;
   (iii) the value of $x$ for which $\sinh^{-1}x=\ln3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions