| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 24 |
| Topic | Hyperbolic functions |
| Type | Reduction formulas with hyperbolic integrals |
| Difficulty | Hard +2.3 This Pre-U Further Maths question requires multiple advanced techniques: proving a reduction formula for hyperbolic integrals with specific limits, computing arc length with complex parametric equations involving hyperbolic functions, and deriving the surface area of a torus using parametric integration. The hyperbolic function manipulations, multi-step proofs, and non-standard parametric forms demand sophisticated problem-solving beyond routine Further Maths exercises. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08d Volumes of revolution: about x and y axes4.08f Integrate using partial fractions8.06a Reduction formulae: establish, use, and evaluate recursively8.06b Arc length and surface area: of revolution, cartesian or parametric |
\begin{enumerate}[label=(\alph*)]
\item Let $I_n = \int_0^{\alpha} \cosh^n x \, dx$ for integers $n \geqslant 0$, where $\alpha = \ln 2$.
\begin{enumerate}[label=(\roman*)]
\item Prove that, for $n \geqslant 2$, $nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}$. [5]
\item A curve has parametric equations $x = 12 \sinh t + 4 \sinh^3 t$, $y = 3 \cosh^4 t$, $0 \leqslant t \leqslant \ln 2$. Find the length of the arc of this curve, giving your answer in the form $a + b \ln 2$ for rational numbers $a$ and $b$. [8]
\end{enumerate}
\item The circle with equation $x^2 + (y - R)^2 = r^2$, where $r < R$, is rotated through one revolution about the $x$-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of $\pi$, $R$ and $r$, the surface area of this torus. [11]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q13 [24]}}