OCR MEI C4 — Question 4 7 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem and Partial Fractions
TypeDirect binomial expansion of quotient
DifficultyStandard +0.3 This is a straightforward application of the binomial expansion requiring students to either use partial fractions or rewrite as (1+2x)(1-2x)^{-2} and multiply series. It's slightly above average difficulty due to the need to handle the product of two expansions and track coefficients carefully, but it's a standard C4 technique with no novel problem-solving required.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
4 Find the first three terms in the binomial expansion of \(\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\frac{1+2x}{(1-2x)^2} = (1+2x)(1-2x)^{-2}\)M1 Binomial expansion power \(-2\)
\(= (1+2x)\left[1+(-2)(-2x)+\frac{(-2)(-3)}{1\cdot2}(-2x)^2+\ldots\right]\)A1 Unsimplified, correct
\(= (1+2x)[1+4x+12x^2+\ldots]\)A1 Sufficient terms
\(= 1+4x+12x^2+2x+8x^2+\ldots\)M1
\(= 1+6x+20x^2+\ldots\)A1
Valid for \(-1 < -2x < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}\)M1
A1
[7] 
## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1+2x}{(1-2x)^2} = (1+2x)(1-2x)^{-2}$ | M1 | Binomial expansion power $-2$ |
| $= (1+2x)\left[1+(-2)(-2x)+\frac{(-2)(-3)}{1\cdot2}(-2x)^2+\ldots\right]$ | A1 | Unsimplified, correct |
| $= (1+2x)[1+4x+12x^2+\ldots]$ | A1 | Sufficient terms |
| $= 1+4x+12x^2+2x+8x^2+\ldots$ | M1 | |
| $= 1+6x+20x^2+\ldots$ | A1 | |
| Valid for $-1 < -2x < 1 \Rightarrow -\frac{1}{2} < x < \frac{1}{2}$ | M1 | |
| | A1 | |
| **[7]** | | |

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4 Find the first three terms in the binomial expansion of $\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }$ 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  Q4 [7]}}
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