| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 7 |
| Topic | Generalised Binomial Theorem |
| Type | Expansion with algebraic manipulation |
| Difficulty | Challenging +1.2 This question requires applying the binomial expansion formula with fractional power (standard technique), identifying validity range (routine), then making a substitution to approximate an integral. The connection between √(4-2x²) and √(cos x) requires insight but is guided by 'hence'. Part (d) tests understanding of validity conditions. Multi-step with some problem-solving but mostly standard A-level Further Maths techniques. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions1.08h Integration by substitution |
12.
\begin{enumerate}[label=(\alph*)]
\item Show that the first two terms of the binomial expansion of $\sqrt { 4 - 2 x ^ { 2 } }$ are
$$2 - \frac { x ^ { 2 } } { 2 }$$
\item State the range of values of $x$ for which the expansion found in part (a) is valid.
\item Hence, find an approximation for
$$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } d x$$
giving your answer to five decimal places.\\
Fully justify your answer.
\item A student decides to use this method to find an approximation for
$$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } d x$$
Explain why this may not be a suitable method.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q12 [7]}}