Core Pure: Hyperbolic functionsEdexcel A-Level Further Maths: Topic test

1. 1
   Let $\mathrm{p}(x)=4\sinh x+3\cosh x$ for $x\in\mathbb{R}$.

   (a)

   What is the value of $\mathrm{p}(0)$?

   [1 mark]

   • A$0$
   • B$3$
   • C$4$
   • D$7$

   (b)

   Which expression is equal to $\mathrm{p}(x)$?

   [1 mark]

   • A$\frac{7e^{x}-e^{-x}}{2}$
   • B$\frac{7e^{-x}-e^{x}}{2}$
   • C$\frac{7e^{x}+e^{-x}}{2}$
   • D$\frac{7e^{x}-7e^{-x}}{2}$

   (c)

   Show that the equation $\mathrm{p}(x)=0$ has exactly one real solution and find it, giving your answer in terms of $\ln7$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=2+\tanh x$.

   (a)

   What is the range of $y$ on $C$?

   [1 mark]

   • A$0<y<2$
   • B$-1<y<1$
   • C$1<y<3$
   • D$y>1$

   (b)

   Which transformation maps $C$ onto itself?

   [1 mark]

   • AReflection in the $y$-axis
   • BReflection in the line $y=2$
   • CRotation through $180^\circ$ about the origin
   • DRotation through $180^\circ$ about the point $(0,2)$

   (c)

   Find the exact value of $x$ at the point on $C$ where $y=2.5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $y=\frac{\cosh2x}{x}$ for $x>0$.

   (a)

   Find $\frac{dy}{dx}$, giving your answer as a single fraction.

   [3 marks]

   (b)

   Find the exact value of $\int_0^{\ln2}\cosh2x\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\cosh2x-5\cosh x$, and $I=\int_0^3\frac{1}{\sqrt{x^2+16}}\,dx$.

   (a)

   Find the exact coordinates of the stationary points of $C$ and determine their nature.

   [6 marks]

   (b)

   Use the substitution $x=4\sinh u$ to show that $I=\ln2$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $y=\ln(\cosh x)$ for $x\in\mathbb{R}$.

   (a)

   What is $\frac{dy}{dx}$?

   [1 mark]

   • A$\frac{1}{\cosh x}$
   • B$\sinh x$
   • C$\frac{\cosh x}{\sinh x}$
   • D$\tanh x$

   (b)

   What is the value of $\frac{d^2y}{dx^2}$ when $x=0$?

   [1 mark]

   • A$0$
   • B$-1$
   • C$1$
   • D$2$

   (c)

   Hence find the exact value of $\int_0^{\ln3}\tanh x\,dx$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $\mathrm{g}(x)=\operatorname{arsinh}\left(\frac{3x}{4}\right)$ for $x\in\mathbb{R}$.

   (a)

   What is the exact value of $\mathrm{g}\left(\frac43\right)$?

   [1 mark]

   • A$\ln2$
   • B$\ln\left(1+\sqrt2\right)$
   • C$\ln\left(\sqrt2-1\right)$
   • D$1$

   (b)

   What is the range of $\mathrm{g}$?

   [1 mark]

   • AAll real numbers
   • B$\mathrm{g}(x)\ge0$
   • C$-1<\mathrm{g}(x)<1$
   • D$\mathrm{g}(x)\ge1$

   (c)

   Solve $\mathrm{g}(x)=\ln2$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the equation $3\sinh x+4\cosh x=6$.

   (a)

   Show that the equation can be written as $7e^{2x}-12e^{x}+1=0$.

   [3 marks]

   (b)

   Hence solve the equation, giving your answers as natural logarithms.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $I=\int_5^9\frac{1}{\sqrt{x^2-9}}\,dx$ and $J=\int_0^{\ln3}x\sinh x\,dx$.

   (a)

   Use the substitution $x=3\cosh u$ to find the exact value of $I$, giving your answer as a single logarithm.

   [6 marks]

   (b)

   Use integration by parts to find the exact value of $J$.

   [6 marks]

   Total for question 8: 12 marks

End of questions