| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Intersection points of hyperbolic curves |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question on hyperbolic functions requiring solving cosh x = sinh 2x using identities (leading to a quadratic in e^x), sketching standard curves, and applying the arc length formula. While it involves several steps and Further Maths content, each part follows standard techniques without requiring novel insight—the equation solving is methodical, sketches are routine, and arc length is a direct formula application with sinh x derivative. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 14.08f Integrate using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\cosh a = 2\sinh a \cosh a \Rightarrow \sinh a = \frac{1}{2}\) | M1 A1 | |
| \(a = \sinh^{-1}\frac{1}{2} = \ln\!\left(\frac{1}{2} + \sqrt{\frac{1}{4}+1}\right)\) | M1 | |
| \(a = \ln\!\left(\frac{1}{2} + \frac{1}{2}\sqrt{5}\right)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graph of \(C_1\) correct | B1 | \(C_1\) correct |
| Graph of \(C_2\) correct and intersecting \(C_1\) in first quadrant | B1 | \(C_2\) correct and intersecting \(C_1\) in first quadrant |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^a \sqrt{1+\sinh^2 x}\,dx\) | M1 | |
| \(\int_0^a \sqrt{\cosh^2 x}\,dx = \int_0^a \cosh x\,dx\) | M1 A1 | |
| \(\left[\sinh x\right]_0^a = \sinh a\) | M1 | |
| \(\frac{1}{2}\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\cosh a = 2\sinh a \cosh a \Rightarrow \sinh a = \frac{1}{2}$ | M1 A1 | |
| $a = \sinh^{-1}\frac{1}{2} = \ln\!\left(\frac{1}{2} + \sqrt{\frac{1}{4}+1}\right)$ | M1 | |
| $a = \ln\!\left(\frac{1}{2} + \frac{1}{2}\sqrt{5}\right)$ | A1 | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graph of $C_1$ correct | B1 | $C_1$ correct |
| Graph of $C_2$ correct and intersecting $C_1$ in first quadrant | B1 | $C_2$ correct and intersecting $C_1$ in first quadrant |
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^a \sqrt{1+\sinh^2 x}\,dx$ | M1 | |
| $\int_0^a \sqrt{\cosh^2 x}\,dx = \int_0^a \cosh x\,dx$ | M1 A1 | |
| $\left[\sinh x\right]_0^a = \sinh a$ | M1 | |
| $\frac{1}{2}$ | A1 | |
---5 The curves $C _ { 1 } : y = \cosh x$ and $C _ { 2 } : y = \sinh 2 x$ intersect at the point where $x = a$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $a$, giving your answer in logarithmic form.
\item Sketch $C _ { 1 }$ and $C _ { 2 }$ on the same diagram.
\item Find the exact value of the length of the arc of $C _ { 1 }$ from $x = 0$ to $\mathrm { x } = \mathrm { a }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q5 [11]}}