OCR MEI C4 — Question 7 6 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeExpand and state validity
DifficultyModerate -0.3 This is a straightforward application of the binomial expansion for fractional powers requiring factoring out the constant, applying the standard formula (1+x)^n with n=1/2, and stating the validity condition |x|<1 after adjustment. It's slightly easier than average as it's a routine textbook exercise with clear steps and no problem-solving insight required, though it does require careful algebraic manipulation.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
7 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.
Question 7:
AnswerMarks Guidance
\(\sqrt{4+2x} = 2(1+\frac{1}{2}x)^{\frac{1}{2}}\)M1 Taking out 4 oe
\(= 2\{1 + \frac{1}{2}\cdot(\frac{1}{2}x) + \frac{\frac{1}{2}\cdot(-\frac{1}{2})}{2!}(\frac{1}{2}x)^2 + \frac{\frac{1}{2}\cdot(-\frac{1}{2})\cdot(-\frac{3}{2})}{3!}(\frac{1}{2}x)^3 + ...\}\)M1 Correct binomial coefficients
\(= k\left(1 + \frac{1}{4}x - \frac{1}{32}x^2 + \frac{1}{128}x^3 + ...\right)\)A2,1,0 \(\frac{1}{4}x, -\frac{1}{32}x^2, +\frac{1}{128}x^3\)
\(= \left(2 + \frac{1}{2}x - \frac{1}{16}x^2 + \frac{1}{64}x^3 + ...\right)\)A1cao
Valid for \(-2 < x < 2\)B1cao
[6]
## Question 7:

$\sqrt{4+2x} = 2(1+\frac{1}{2}x)^{\frac{1}{2}}$ | **M1** | Taking out 4 oe

$= 2\{1 + \frac{1}{2}\cdot(\frac{1}{2}x) + \frac{\frac{1}{2}\cdot(-\frac{1}{2})}{2!}(\frac{1}{2}x)^2 + \frac{\frac{1}{2}\cdot(-\frac{1}{2})\cdot(-\frac{3}{2})}{3!}(\frac{1}{2}x)^3 + ...\}$ | **M1** | Correct binomial coefficients

$= k\left(1 + \frac{1}{4}x - \frac{1}{32}x^2 + \frac{1}{128}x^3 + ...\right)$ | **A2,1,0** | $\frac{1}{4}x, -\frac{1}{32}x^2, +\frac{1}{128}x^3$

$= \left(2 + \frac{1}{2}x - \frac{1}{16}x^2 + \frac{1}{64}x^3 + ...\right)$ | **A1cao** |

Valid for $-2 < x < 2$ | **B1cao** |

**[6]**
7 Find the first 4 terms in the binomial expansion of $\sqrt { 4 + 2 x }$. State the range of values of $x$ for which the expansion is valid.

\hfill \mbox{\textit{OCR MEI C4  Q7 [6]}}
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