CAIE P3 2013 June — Question 2 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2013
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeProduct with linear term
DifficultyModerate -0.3 This is a straightforward application of the binomial expansion requiring students to expand (1+2x)^(-1/2) using the standard formula, then multiply by (1+3x) and collect terms up to x². While it involves fractional powers and algebraic manipulation, it's a routine textbook exercise with no conceptual challenges beyond applying the memorised formula correctly.
Spec1.04c Extend binomial expansion: rational n, |x|<1
2 Expand \(\frac { 1 + 3 x } { \sqrt { } ( 1 + 2 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
Question 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Obtain \(1 - x\) as first two terms of \((1+2x)^{-\frac{1}{2}}\)B1
Obtain \(+\frac{3}{2}x^2\) or unsimplified equivalent as third term of \((1+2x)^{-\frac{1}{2}}\)B1
Multiply \(1+3x\) by attempt at \((1+2x)^{-\frac{1}{2}}\), obtaining sufficient termsM1
Obtain final answer \(1 + 2x - \frac{3}{2}x^2\)A1 [4] 
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $1 - x$ as first two terms of $(1+2x)^{-\frac{1}{2}}$ | B1 | |
| Obtain $+\frac{3}{2}x^2$ or unsimplified equivalent as third term of $(1+2x)^{-\frac{1}{2}}$ | B1 | |
| Multiply $1+3x$ by attempt at $(1+2x)^{-\frac{1}{2}}$, obtaining sufficient terms | M1 | |
| Obtain final answer $1 + 2x - \frac{3}{2}x^2$ | A1 | [4] |

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2 Expand $\frac { 1 + 3 x } { \sqrt { } ( 1 + 2 x ) }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$, simplifying the coefficients.

\hfill \mbox{\textit{CAIE P3 2013 Q2 [4]}}
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