FP3: Hyperbolic functionsEdexcel International A Level Further Maths: Topic test

1. 1
   The real number $x$ satisfies $\sinh x=\frac34$.

   (a)

   Find $\cosh x$.

   [1 mark]

   • A$\frac{\sqrt7}{4}$
   • B$\frac54$
   • C$\frac{25}{16}$
   • D$\frac74$

   (b)

   Find $\tanh x$.

   [1 mark]

   • A$\frac43$
   • B$\frac{15}{16}$
   • C$\frac34$
   • D$\frac35$

   (c)

   Find the exact value of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The real numbers $a$ and $b$ are defined by $a=\operatorname{arsinh}\frac{5}{12}$ and $b=\operatorname{arcosh}\frac53$.

   (a)

   Find $a$ in exact logarithmic form.

   [1 mark]

   • A$\ln\frac32$
   • B$\ln\frac23$
   • C$\ln\frac{13}{12}$
   • D$\ln\frac{17}{12}$

   (b)

   Find $b$ in exact logarithmic form.

   [1 mark]

   • A$\ln\frac53$
   • B$\ln\frac43$
   • C$\ln3$
   • D$\ln\frac13$

   (c)

   Find the exact value of $\sinh(a+b)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $\cosh2x-3\sinh x=1$.

   (a)

   Show that the equation can be written as $2\sinh^2x-3\sinh x=0$.

   [3 marks]

   (b)

   Hence solve the equation, giving your non-zero answer in exact logarithmic form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=3\cosh x-\sinh x$ and has exactly one stationary point, which is a minimum.

   (a)

   Find the exact coordinates of the minimum point of $C$.

   [6 marks]

   (b)

   Solve the equation $3\cosh x-\sinh x=4$, giving your answers as exact logarithms.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The real number $x$ satisfies $\cosh x=\frac{13}{5}$ and $x>0$.

   (a)

   Find $\sinh x$.

   [1 mark]

   • A$\frac{\sqrt{194}}{5}$
   • B$\frac{5}{12}$
   • C$\frac{12}{5}$
   • D$\frac{169}{25}$

   (b)

   Find $\cosh2x$.

   [1 mark]

   • A$\frac{313}{25}$
   • B$\frac{338}{25}$
   • C$\frac{288}{25}$
   • D$\frac{312}{25}$

   (c)

   Find the exact value of $\tanh2x$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The real number $p$ is defined by $p=\operatorname{arsinh}2$.

   (a)

   Find $p$ in exact logarithmic form.

   [1 mark]

   • A$\ln\left(2+\sqrt3\right)$
   • B$\ln\left(\sqrt5-2\right)$
   • C$\ln\left(1+\sqrt5\right)$
   • D$\ln\left(2+\sqrt5\right)$

   (b)

   Find $\cosh p$.

   [1 mark]

   • A$5$
   • B$\sqrt5$
   • C$\sqrt3$
   • D$\frac{1}{\sqrt5}$

   (c)

   Show that $\mathrm{e}^p-\mathrm{e}^{-p}=4$, using the logarithmic form of $p$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   This question is about the double-angle result $\sinh2x\equiv2\sinh x\cosh x$.

   (a)

   Prove that $\sinh2x\equiv2\sinh x\cosh x$, using the definitions of $\sinh$ and $\cosh$ in terms of exponentials.

   [3 marks]

   (b)

   Hence solve the equation $\sinh2x=5\sinh x$, giving your answers in exact form.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   For $x\ge1$, let $y=\operatorname{arcosh}x$, where $y\ge0$.

   (a)

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

   [6 marks]

   (b)

   Use the result of part (a), and the logarithmic form of $\operatorname{arsinh}$, to solve the equation $\operatorname{arcosh}x=2\operatorname{arsinh}\frac{8}{15}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions