Hyperbolic functions and their graphsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $x=\ln3$.

   (a)

   Find the value of $\sinh x$.

   [1 mark]

   • A$\frac53$
   • B$\frac83$
   • C$\frac43$
   • D$\frac45$

   (b)

   Find the value of $\tanh x$.

   [1 mark]

   • A$\frac54$
   • B$\frac45$
   • C$\frac43$
   • D$\frac53$

   (c)

   Hence, or otherwise, find the value of $\cosh x+\sinh x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The functions $f$, $g$ and $h$ are defined for all real $x$ by $f(x)=\sinh x$, $g(x)=\cosh x$ and $h(x)=\tanh x$.

   (a)

   Which gives the range of $h$?

   [1 mark]

   • A$-1<h(x)<1$
   • B$h(x)\ge0$
   • C$-1\le h(x)\le1$
   • D$h(x)\in\mathbb{R}$

   (b)

   Which statement is true?

   [1 mark]

   • A$g$ is an odd function
   • B$f$ is an even function
   • C$h(x)\to\infty$ as $x\to\infty$
   • D$g(-x)=g(x)$ and $f(-x)=-f(x)$

   (c)

   Explain why the equation $g(x)=\frac12$ has no real solutions.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $\cosh x=\frac{13}{12}$ has two real solutions.

   (a)

   Use the definition of $\cosh x$ to show that $6e^{2x}-13e^x+6=0$.

   [3 marks]

   (b)

   Hence find the two solutions of the equation, giving your answers in exact form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cable hangs between two poles. The height of the cable above the ground, $y$ metres, at horizontal distance $x$ metres from the lowest point of the cable is $y=4\cosh\left(\frac{x}{4}\right)$. The cable is attached to the top of each of two poles, which stand at $x=-8$ and $x=8$.

   (a)

   Show that the height of the cable is $5$ m when $x=\pm4\ln2$.

   [6 marks]

   (b)

   (i) Explain why the two poles have the same height.
   (ii) Find the exact height of each pole.
   (iii) Find, to $3$ significant figures, how much higher the top of each pole is than the lowest point of the cable.

   [6 marks]

   Total for question 4: 12 marks

End of questions