Hyperbolic functionsAQA A-Level Further Maths: Topic test

1. 1
   Let $x=\ln5$.

   (a)

   What is the exact value of $\sinh x$?

   [1 mark]

   • A$\frac{13}{5}$
   • B$\frac{12}{5}$
   • C$\frac{24}{5}$
   • D$2$

   (b)

   What is the exact value of $\tanh x$?

   [1 mark]

   • A$\frac{24}{25}$
   • B$\frac{13}{12}$
   • C$\frac{12}{5}$
   • D$\frac{12}{13}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $F$ is defined by $F(x)=\operatorname{arcosh}(2x+1)$ for the largest possible real domain.

   (a)

   What is the domain of $F$?

   [1 mark]

   • A$x\ge0$
   • B$x\ge1$
   • C$x\ge-\frac12$
   • Dall real $x$

   (b)

   What is the range of $F$?

   [1 mark]

   • Aall real numbers
   • B$F(x)\ge1$
   • C$F(x)\ge0$
   • D$F(x)>0$

   (c)

   Solve $F(x)=\ln3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $y=\operatorname{artanh}x$ for $-1<x<1$.

   (a)

   Show that $y=\frac12\ln\left(\frac{1+x}{1-x}\right)$.

   [3 marks]

   (b)

   Given that $\operatorname{artanh}x-\operatorname{artanh}\left(\frac15\right)=\ln2$, find the value of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=5\cosh x-4\sinh x$ for all real $x$.

   (a)

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

   [6 marks]

   (b)

   Find the exact area of the region bounded by $C$, the $x$-axis, the $y$-axis and the line $x=\ln3$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A real number $x>0$ satisfies $\cosh x=\dfrac{17}{8}$.

   (a)

   What is the value of $\sinh x$?

   [1 mark]

   • A$\frac98$
   • B$\frac{225}{64}$
   • C$\frac{15}{8}$
   • D$\frac{\sqrt{353}}{8}$

   (b)

   What is the value of $\tanh x$?

   [1 mark]

   • A$\frac{15}{17}$
   • B$\frac{17}{15}$
   • C$\frac{15}{8}$
   • D$\frac{8}{17}$

   (c)

   Find the exact value of $x$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $t=\tanh x$, where $x$ is a real number.

   (a)

   Which expression is equal to $\operatorname{sech}^2x$?

   [1 mark]

   • A$1+t^2$
   • B$t^2-1$
   • C$\frac{1}{1-t^2}$
   • D$1-t^2$

   (b)

   Which expression is equal to $\cosh2x$?

   [1 mark]

   • A$\frac{1-t^2}{1+t^2}$
   • B$\frac{1+t^2}{1-t^2}$
   • C$1+t^2$
   • D$\frac{2t}{1+t^2}$

   (c)

   Given that $t=\frac12$, find the exact value of $\operatorname{sech}x$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has equation $y=\ln(\cosh x)$ for all real $x$.

   (a)

   Show that $\frac{dy}{dx}=\tanh x$, and show that $C$ has no point of inflection.

   [3 marks]

   (b)

   Hence find the exact value of $\int_0^{\ln2}\tanh x\,dx$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The integral $J$ is defined by $J=\displaystyle\int_{-2}^{1}\dfrac{1}{\sqrt{x^2+4x+13}}\,dx$.

   (a)

   Find the exact value of $J$, giving your answer as a single natural logarithm.

   [6 marks]

   (b)

   Independently of part (a), prove that $\cosh3x=4\cosh^3x-3\cosh x$, and hence find the exact value of $\cosh(3\ln2)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions