| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 16 |
| Topic | Taylor series |
| Type | Use series to approximate numerical value |
| Difficulty | Challenging +1.2 This question requires knowledge of Maclaurin series for sinh x and sin x, but the execution is relatively straightforward: showing a root lies in an interval by substitution, then using series truncation to solve polynomial approximations. The algebraic manipulation (solving x^3 terms, then x^5 terms) is standard for Further Maths students. While it involves multiple steps and series work, it follows a predictable pattern without requiring novel insight or particularly complex reasoning. |
| Spec | 1.09a Sign change methods: locate roots4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
**Question 6** [2+3+3 marks]
**(i)**
For $f(x) = \sinh x + \sin x - 3x$,
$f(2.5) = -0.851\ldots < 0$ and $f(3) = 1.159\ldots > 0$ M1 (or LHS < RHS and then LHS > RHS)
Change-of-sign (for a continuous fn.) $\Rightarrow 2.5 < \alpha < 3$ A1 (All correctly shown/explained)
**[2]**
**(ii)**
$\sinh x + \sin x = \left(x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \frac{x^9}{9!}+\ldots\right) + \left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!}-\ldots\right)$ M1 (for use of both series (attempted))
$= 2x + \frac{x^5}{60} + \ldots$ A1
$2x + \frac{x^5}{60} = 3x \Rightarrow (x\neq 0)\ x^4 = 60$
$\Rightarrow \alpha \approx \sqrt[4]{60}\ (2.783\,158\ldots)$ B1 **(AG)** shown legitimately
**[3]**
**(iii)**
Using $2x + \frac{x^5}{60} + \frac{x^9}{181\,440} = 3x$ with $x \neq 0$ M1
Solving as a quadratic in $x^4$ M1 ($x^8 + 3024x^4 - 181\,440 = 0$)
$\alpha \approx 2.769\,8$ (to 4 d.p.) A1 (from $x^4 = \sqrt{2\,467\,584}-1512$, $x = \sqrt[4]{58.854\,5\ldots}$)
[c.f. actual root 2.769 7 to 4 d.p.]
**[3]**6 The equation $\sinh x + \sin x = 3 x$ has one positive root $\alpha$.\\
(i) Show that $2.5 < \alpha < 3$.\\
(ii) By using the first two non-zero terms in the Maclaurin series for $\sinh x + \sin x$, show that $\alpha \approx \sqrt [ 4 ] { 60 }$.\\
(iii) By taking the third non-zero term in this series, find a second approximation to $\alpha$, giving your answer correct to 4 decimal places.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q6 [16]}}