OCR MEI C4 — Question 3 5 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeExpand and state validity
DifficultyModerate -0.5 This is a straightforward application of the binomial expansion formula requiring factoring out constants and using (1+x)^n with n=-1. While it requires careful algebraic manipulation and understanding of validity conditions (|3x/2| < 1), it's a standard textbook exercise with no novel problem-solving required, making it slightly easier than average.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
3 Find the first three terms of the binomial expansion of \(\frac { 1 } { 2 - 3 x }\).
Give the range of values of \(x\) for which the expansion is valid.
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(\frac{1}{2-3x} = \frac{1}{2}\left(1 - \frac{3}{2}x\right)^{-1}\)M1 Extracting \(\frac{1}{2}\)
\(= \frac{1}{2}\left(1 + (-1)\left(-\frac{3}{2}x\right) + \frac{(-1)(-2)}{1\times 2}\left(-\frac{3}{2}x\right)^2 + \ldots\right)\)A1 M1
\(= \frac{1}{2}\left(1 + \frac{3}{2}x + \frac{9}{4}x^2 + \ldots\right)\)
\(= \frac{1}{2} + \frac{3}{4}x + \frac{9}{8}x^2\)A1
Valid if \(-1 < \frac{3}{2}x < 1 \Rightarrow -\frac{2}{3} < x < \frac{2}{3}\)B1
Total: 5 marks 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2-3x} = \frac{1}{2}\left(1 - \frac{3}{2}x\right)^{-1}$ | M1 | Extracting $\frac{1}{2}$ |
| $= \frac{1}{2}\left(1 + (-1)\left(-\frac{3}{2}x\right) + \frac{(-1)(-2)}{1\times 2}\left(-\frac{3}{2}x\right)^2 + \ldots\right)$ | A1 M1 | |
| $= \frac{1}{2}\left(1 + \frac{3}{2}x + \frac{9}{4}x^2 + \ldots\right)$ | | |
| $= \frac{1}{2} + \frac{3}{4}x + \frac{9}{8}x^2$ | A1 | |
| Valid if $-1 < \frac{3}{2}x < 1 \Rightarrow -\frac{2}{3} < x < \frac{2}{3}$ | B1 | |
| **Total: 5 marks** | | |

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3 Find the first three terms of the binomial expansion of $\frac { 1 } { 2 - 3 x }$.\\
Give the range of values of $x$ for which the expansion is valid.

\hfill \mbox{\textit{OCR MEI C4  Q3 [5]}}
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