| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Generalised Binomial Theorem |
| Type | Product of separate expansions |
| Difficulty | Standard +0.3 This is a standard binomial expansion question requiring routine application of the generalised binomial theorem with fractional/negative indices. Part (i) is trivial recall, part (ii) requires factoring out constants and applying the formula, and part (iii) involves multiplying two expansions and basic integration—all mechanical procedures with no novel insight required. Slightly above average difficulty due to the multi-step nature and integration application, but remains a textbook exercise. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions1.08d Evaluate definite integrals: between limits |
11
\begin{enumerate}[label=(\roman*)]
\item Expand $( 1 + x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
\item (a) Expand $\sqrt { 2 + 3 x ^ { 2 } }$ up to and including the term in $x ^ { 4 }$.\\
(b) For what range of values of $x$ is this expansion valid?
\item Find the first three terms of the expansion of $\frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x }$ in ascending powers of $x$ and hence show that $\int _ { 0 } ^ { 0.1 } \frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x } \mathrm {~d} x \approx 0.135$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2013 Q11}}