| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Use series to approximate integral |
| Difficulty | Standard +0.3 This is a straightforward application of standard Maclaurin series for sinh and cosh with simple substitution (x²), followed by routine differentiation coefficient extraction and polynomial integration. All steps are mechanical with no problem-solving insight required, though it's slightly above average difficulty due to being Further Maths content. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) | ( ) ( ) | |
| sinh x2 +cosh x2 | M1 | Combines using correct power series. Or sensible |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks |
|---|---|
| 1(b) | 1×4!=12 |
| 2 | B1 |
| Answer | Marks |
|---|---|
| 1(c) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 0 | M1 | Substitutes their power series, must be at least a+bx2. |
| Answer | Marks |
|---|---|
| 3 10 0 960 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 1:
--- 1(a) ---
1(a) | ( ) ( )
sinh x2 +cosh x2 | M1 | Combines using correct power series. Or sensible
attempt at all four derivatives of y.
1+x2 +1x4
2 | A1
2
--- 1(b) ---
1(b) | 1×4!=12
2 | B1
1
--- 1(c) ---
1(c) | 1
21+x2 + 1x4 dx
2
0 | M1 | Substitutes their power series, must be at least a+bx2.
1
=x+1x3 + 1 x52 = 523
3 10 0 960 | A1
2
Question | Answer | Marks | GuidanceIt is given that $y = \sinh(x^2) + \cosh(x^2)$.
\begin{enumerate}[label=(\alph*)]
\item Use standard results from the list of formulae (MF19) to find the Maclaurin's series for $y$ in terms of $x$ up to and including the term in $x^4$. [2]
\item Deduce the value of $\frac{d^4y}{dx^4}$ when $x = 0$. [1]
\item Use your answer to part (a) to find an approximation to $\int_0^{\frac{1}{2}} y \, dx$, giving your answer as a rational fraction in its lowest terms. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2021 Q1 [5]}}