OCR MEI C4 — Question 5 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 formula for fractional powers with standard validity determination. It requires routine substitution into the formula (1+x)^n = 1 + nx + n(n-1)x²/2! + ..., then finding one more term and stating |2x| < 1. The question is slightly easier than average because it's a direct template exercise with no problem-solving element, though the fractional power and validity requirement keep it from being trivial.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
5 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.
Question 5(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((1+2x)^{1/3} = 1 + \frac{1}{3}\cdot 2x + \frac{\frac{1}{3}\cdot\left(-\frac{2}{3}\right)}{2!}(2x)^2+\ldots\)M1 Binomial expansion
\(= 1+\frac{2}{3}x - \frac{2}{18}\cdot 4x^2+\ldots\)A1 Correct unsimplified expression
\(= 1+\frac{2}{3}x - \frac{4}{9}x^2+\ldots\) *E1 www (simplification)
Next term \(= \frac{\frac{1}{3}\cdot\left(-\frac{2}{3}\right)\left(-\frac{5}{3}\right)}{3!}(2x)^3\)M1
\(= \frac{40}{81}x^3\)A1
Valid for \(-1 < 2x < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}\)B1
[6] 
## Question 5(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1+2x)^{1/3} = 1 + \frac{1}{3}\cdot 2x + \frac{\frac{1}{3}\cdot\left(-\frac{2}{3}\right)}{2!}(2x)^2+\ldots$ | M1 | Binomial expansion |
| $= 1+\frac{2}{3}x - \frac{2}{18}\cdot 4x^2+\ldots$ | A1 | Correct unsimplified expression |
| $= 1+\frac{2}{3}x - \frac{4}{9}x^2+\ldots$ * | E1 | www (simplification) |
| Next term $= \frac{\frac{1}{3}\cdot\left(-\frac{2}{3}\right)\left(-\frac{5}{3}\right)}{3!}(2x)^3$ | M1 | |
| $= \frac{40}{81}x^3$ | A1 | |
| Valid for $-1 < 2x < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}$ | B1 | |
| **[6]** | | |

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5 Show that $( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots$, and find the next term in the expansion.\\
State the set of values of $x$ for which the expansion is valid.

\hfill \mbox{\textit{OCR MEI C4  Q5 [6]}}
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