OCR MEI C4 — Question 3 7 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeExpand and state validity
DifficultyStandard +0.3 This is a straightforward application of the binomial expansion for negative/fractional powers requiring factoring out constants, expanding (1-u)^{-3}, and stating the validity condition |u|<1. It's slightly above average difficulty due to the negative power and need to handle the constant factor correctly, but follows a standard template with no novel problem-solving required.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
3 Find the first three terms in the binomial expansion of \(\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]
Question 3:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(3^{-3} = \frac{1}{3}\cdot 3\cdot 2x^{-3}\left(1-\frac{2}{3}x\right)^{-3}\)M1 Dealing with the '3'
\(= \frac{1}{27}\left(1+(-3)\left(-\frac{2}{3}x\right)+\frac{(-3)(-4)}{2}\left(-\frac{2}{3}x\right)^2+\ldots\right)\)B1 Correct binomial coefficients
\(= \frac{1}{27}\left(1+2x+\frac{8}{3}x^2+\ldots\right)\)B2,1,0 \(1,\ 2,\ 8/3\) oe
\(= \frac{1}{27}+\frac{2}{27}x+\frac{8}{81}x^2+\ldots\)A1 cao
Valid for \(-1 < -\frac{2}{3}x < 1\)M1
\(\Rightarrow -\frac{3}{2} < x < \frac{3}{2}\)A1
[7] 
## Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $3^{-3} = \frac{1}{3}\cdot 3\cdot 2x^{-3}\left(1-\frac{2}{3}x\right)^{-3}$ | M1 | Dealing with the '3' |
| $= \frac{1}{27}\left(1+(-3)\left(-\frac{2}{3}x\right)+\frac{(-3)(-4)}{2}\left(-\frac{2}{3}x\right)^2+\ldots\right)$ | B1 | Correct binomial coefficients |
| $= \frac{1}{27}\left(1+2x+\frac{8}{3}x^2+\ldots\right)$ | B2,1,0 | $1,\ 2,\ 8/3$ oe |
| $= \frac{1}{27}+\frac{2}{27}x+\frac{8}{81}x^2+\ldots$ | A1 | cao |
| Valid for $-1 < -\frac{2}{3}x < 1$ | M1 | |
| $\Rightarrow -\frac{3}{2} < x < \frac{3}{2}$ | A1 | |
| **[7]** | | |

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3 Find the first three terms in the binomial expansion of $\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }$ in ascending powers of $x$. State the set of values of $x$ for which the expansion is valid.\\[0pt]
[7]

\hfill \mbox{\textit{OCR MEI C4  Q3 [7]}}
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