CAIE P1 2016 November — Question 4 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2016
SessionNovember
Marks6
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Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeTwo equations from coefficients
DifficultyStandard +0.3 This is a straightforward binomial expansion problem requiring students to expand (1 + x/2)^n, multiply by (3 - 2x), equate the coefficient of x to 7 to find n, then find the x² coefficient. It involves standard algebraic manipulation and binomial coefficient calculation, making it slightly easier than average as it follows a predictable template with no novel insight required.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
4 In the expansion of \(( 3 - 2 x ) \left( 1 + \frac { x } { 2 } \right) ^ { n }\), the coefficient of \(x\) is 7 . Find the value of the constant \(n\) and hence find the coefficient of \(x ^ { 2 }\).
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Term in \(x = \frac{nx}{2}\)B1 Could be implied by use of a numerical \(n\)
\((3-2x)(1 + \frac{nx}{2} + \ldots) \rightarrow 7 = \frac{3n}{2} - 2 \rightarrow n = 6\)M1, A1 Their 2 terms in \(x = 7\)
Term in \(x^2 = \frac{n(n-1)}{2}\left(\frac{x}{2}\right)^2\)B1 May be implied by (their \(n\)) \(\times\) (their \(n-1\)) \(\div 8\)
Coefficient of \(x^2 = \frac{3n(n-1)}{8} - \frac{2n}{2} = \frac{21}{4}\)M1, A1 Considers 2 terms in \(x^2\), aef 
## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Term in $x = \frac{nx}{2}$ | B1 | Could be implied by use of a numerical $n$ |
| $(3-2x)(1 + \frac{nx}{2} + \ldots) \rightarrow 7 = \frac{3n}{2} - 2 \rightarrow n = 6$ | M1, A1 | Their 2 terms in $x = 7$ |
| Term in $x^2 = \frac{n(n-1)}{2}\left(\frac{x}{2}\right)^2$ | B1 | May be implied by (their $n$) $\times$ (their $n-1$) $\div 8$ |
| Coefficient of $x^2 = \frac{3n(n-1)}{8} - \frac{2n}{2} = \frac{21}{4}$ | M1, A1 | Considers 2 terms in $x^2$, aef |

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4 In the expansion of $( 3 - 2 x ) \left( 1 + \frac { x } { 2 } \right) ^ { n }$, the coefficient of $x$ is 7 . Find the value of the constant $n$ and hence find the coefficient of $x ^ { 2 }$.

\hfill \mbox{\textit{CAIE P1 2016 Q4 [6]}}
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