| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 7 |
| Topic | Generalised Binomial Theorem |
| Type | Factoring out constants first |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion with factoring out constants, followed by routine validity determination and a simple substitution. The steps are mechanical: rewrite as 5(4-3x)^(-1/2) = 5ยท2^(-1)(1-3x/4)^(-1/2), apply the standard binomial formula for negative/fractional powers, find validity from |3x/4|<1, and substitute. All techniques are standard A-level pure maths with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
5.
$$f ( x ) = \frac { 10 } { \sqrt { 4 - 3 x } }$$
\begin{enumerate}[label=(\alph*)]
\item Show that the first 4 terms in the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, are
$$A + B x + C x ^ { 2 } + \frac { 675 } { 1024 } x ^ { 3 }$$
where $A , B$ and $C$ are constants to be found. Give each constant in simplest form.
Given that this expansion is valid for $| x | < k$
\item state the largest value of $k$.
By substituting $x = \frac { 1 } { 3 }$ into $\mathrm { f } ( x )$ and into the answer for part (a),
\item find an approximation for $\sqrt { 3 }$
Give your answer in the form $\frac { a } { b }$ where $a$ and $b$ are integers to be found.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q5 [7]}}