OCR MEI C4 — Question 6 5 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeFactoring out constants first
DifficultyModerate -0.8 This is a straightforward application of the binomial expansion formula requiring factoring out the constant 4, then expanding (1 + x/4)^(1/2) using the standard formula. The validity condition |x/4| < 1 is routine. It's easier than average as it's a direct textbook exercise with no problem-solving element, though it does require careful algebraic manipulation.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
6 Find the first three terms in the binomial expansion of \(\overline { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.
Question 6:
AnswerMarks Guidance
AnswerMarks Guidance
\(\sqrt{4+x} = 2\left(1+\frac{x}{4}\right)^{\frac{1}{2}}\)M1 Dealing with \(\sqrt{4}\) (or terms in \(4^{\frac{1}{2}}, 4^{-\frac{1}{2}},\ldots\) etc)
\(= 2\left(1 + \frac{1}{2}\cdot\frac{x}{4} + \frac{\frac{1}{2}\cdot\frac{-1}{2}}{2}\left(\frac{x}{4}\right)^2 + \ldots\right)\)M1 Correct binomial coefficients
\(= 2\left(1 + \frac{1}{8}x - \frac{1}{128}x^2 + \ldots\right)\)A1 Correct unsimplified expression for \((1+x/4)^{\frac{1}{2}}\) or \((4+x)^{\frac{1}{2}}\)
\(= 2 + \frac{1}{4}x - \frac{1}{64}x^2 + \ldots\)A1 cao
Valid for \(-1 < x/4 < 1 \Rightarrow -4 < x < 4\)B1 
## Question 6:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{4+x} = 2\left(1+\frac{x}{4}\right)^{\frac{1}{2}}$ | M1 | Dealing with $\sqrt{4}$ (or terms in $4^{\frac{1}{2}}, 4^{-\frac{1}{2}},\ldots$ etc) |
| $= 2\left(1 + \frac{1}{2}\cdot\frac{x}{4} + \frac{\frac{1}{2}\cdot\frac{-1}{2}}{2}\left(\frac{x}{4}\right)^2 + \ldots\right)$ | M1 | Correct binomial coefficients |
| $= 2\left(1 + \frac{1}{8}x - \frac{1}{128}x^2 + \ldots\right)$ | A1 | Correct unsimplified expression for $(1+x/4)^{\frac{1}{2}}$ or $(4+x)^{\frac{1}{2}}$ |
| $= 2 + \frac{1}{4}x - \frac{1}{64}x^2 + \ldots$ | A1 | cao |
| Valid for $-1 < x/4 < 1 \Rightarrow -4 < x < 4$ | B1 | |

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6 Find the first three terms in the binomial expansion of $\overline { 4 + x }$ in ascending powers of $x$.\\
State the set of values of $x$ for which the expansion is valid.

\hfill \mbox{\textit{OCR MEI C4  Q6 [5]}}
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