Moderate -0.5 This is a straightforward application of the binomial expansion for (1+x)^n with n=1/2, requiring recall of the formula and basic algebraic manipulation. The validity condition (|2x|<1) follows directly from the standard convergence criterion. It's slightly easier than average as it's a routine textbook exercise with no problem-solving element.
Do not MR for \(n \neq \frac{1}{2}\). All four correct binomial coefficients (not nCr form) soi. Accept unsimplified coefficients if a subsequent error when simplifying
\(= 1 + x\)
B1
Condone absence of brackets only if followed by correct work, eg \(2x^2 = 4x^2\) must be soi for second B mark. \(1+x\) www
\(\ldots - \frac{1}{2}x^2\)
B1
\(\ldots - \frac{1}{2}x^2\) www
\(\ldots + \frac{1}{2}x^3 + \ldots\)
B1
\(\ldots + \frac{1}{2}x^3\) www. If there is an error in say the third coefficient of the expansion, M0, B1, B0, B1 can be scored
Valid for \(\
x\
< \frac{1}{2}\) or \(-\frac{1}{2} < x < \frac{1}{2}\)
[5 marks total]
# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1+2x)^{\frac{1}{2}} = 1 + \frac{1}{2}(2x) + \frac{\frac{1}{2}\cdot(-\frac{1}{2})}{2!}(2x)^2 + \frac{\frac{1}{2}\cdot(-\frac{1}{2})\cdot(-\frac{3}{2})}{3!}(2x)^3 + \ldots$ | M1 | Do not MR for $n \neq \frac{1}{2}$. All four correct binomial coefficients (not nCr form) soi. Accept unsimplified coefficients if a subsequent error when simplifying |
| $= 1 + x$ | B1 | Condone absence of brackets only if followed by correct work, eg $2x^2 = 4x^2$ must be soi for second B mark. $1+x$ www |
| $\ldots - \frac{1}{2}x^2$ | B1 | $\ldots - \frac{1}{2}x^2$ www |
| $\ldots + \frac{1}{2}x^3 + \ldots$ | B1 | $\ldots + \frac{1}{2}x^3$ www. If there is an error in say the third coefficient of the expansion, M0, B1, B0, B1 can be scored |
| Valid for $\|x\| < \frac{1}{2}$ or $-\frac{1}{2} < x < \frac{1}{2}$ | B1 | Independent of expansion. $\|x\| \leq \frac{1}{2}$ and $-\frac{1}{2} \leq x \leq \frac{1}{2}$ are actually correct so will be accepted. Condone a combination of inequalities. Condone $-\frac{1}{2} < \|x\| < \frac{1}{2}$ but not $x < \frac{1}{2}$ or $-1 < 2x < 1$ or $-\frac{1}{2} > x > \frac{1}{2}$ |
**[5 marks total]**
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4 Find the first four terms in the binomial expansion of $\sqrt { 1 + 2 x }$. State the set of values of $x$ for which the expansion is valid.
\hfill \mbox{\textit{OCR MEI C4 Q4 [5]}}