Hyperbolic functions and identitiesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A student evaluates hyperbolic functions at $x=\ln3$, using $\sinh x=\frac{e^x-e^{-x}}{2}$ and $\cosh x=\frac{e^x+e^{-x}}{2}$.

   (a)

   Find the value of $\cosh(\ln3)$.

   [1 mark]

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

   (b)

   Find the value of $\tanh(\ln3)$.

   [1 mark]

   • A$\frac45$
   • B$\frac54$
   • C$\frac35$
   • D$\frac43$

   (c)

   Use the identity $\cosh2x=\cosh^2x+\sinh^2x$ to find the exact value of $\cosh(2\ln3)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=\tanh x$ for all real $x$.

   (a)

   What is the range of $f$?

   [1 mark]

   • A$f(x)\ge1$
   • B$-1<f(x)<1$
   • C$f(x)\in\mathbb{R}$
   • D$0<f(x)<1$

   (b)

   Which of the following is equal to $f(-x)$?

   [1 mark]

   • A$f(x)$
   • B$\frac{1}{f(x)}$
   • C$-\frac{1}{f(x)}$
   • D$-f(x)$

   (c)

   Show that $\tanh x=\frac{e^{2x}-1}{e^{2x}+1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For real $x$, $\sinh x=\frac{e^x-e^{-x}}{2}$ and $\cosh x=\frac{e^x+e^{-x}}{2}$.

   (a)

   Prove that $\cosh^2x-\sinh^2x=1$.

   [3 marks]

   (b)

   Using the result in part (a) and the identity $\cosh2x=\cosh^2x+\sinh^2x$, solve $\cosh2x+3\sinh x=3$. Give your answers to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x)=5\cosh x+3\sinh x$ for all real $x$.

   (a)

   Show that the equation $f(x)=5$ can be written as $4e^{2x}-5e^x+1=0$, and hence solve it, giving exact answers.

   [6 marks]

   (b)

   Show that $f(x)-4=\left(2e^{x/2}-e^{-x/2}\right)^2$. Hence find the minimum value of $f(x)$ and the exact value of $x$ at which it occurs.

   [6 marks]

   Total for question 4: 12 marks

End of questions