Moderate -0.5 This is a straightforward application of the binomial expansion formula for fractional powers with standard validity condition |2x| < 1. It requires direct substitution into the formula and simplification, making it slightly easier than average as it's a routine textbook exercise with no problem-solving element.
2 Find the first four terms of the binomial expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\).
State the set of values of \(x\) for which the expansion is valid.
## Question 2:
$(1-2x)^{\frac{1}{2}} \approx 1 + \frac{\frac{1}{2}\left(-\frac{1}{2}\right)}{2!}(-2x)^2 + \frac{\frac{1}{2}\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{3!}(-2x)^3$ | M1 | AO 1.1 | Binomial coefficients seen, allow one error
$= 1 - x - \frac{1}{2}x^2 - \frac{1}{2}x^3$ | A2 | AO 1.1, 1.1 | $1-x$, $-\frac{1}{2}x^2$, $-\frac{1}{2}x^3$ or A1 for 2 correct terms
Valid for $-\frac{1}{2} < x < \frac{1}{2}$ | B1 | AO 2.3 | or $|x| < \frac{1}{2}$; series converges for $x = \pm\frac{1}{2}$; candidates not expected to know this but allow $\leq$ for either or both inequalities
**[4 marks]**
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2 Find the first four terms of the binomial expansion of $( 1 - 2 x ) ^ { \frac { 1 } { 2 } }$.
State the set of values of $x$ for which the expansion is valid.
\hfill \mbox{\textit{OCR MEI Paper 3 Q2 [4]}}