CAIE P1 2014 November — Question 1 3 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionNovember
Marks3
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Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeRatio of coefficients condition
DifficultyModerate -0.8 This is a straightforward application of the binomial theorem requiring students to write out two coefficient terms, set them equal, and solve a simple linear equation for a. The question involves routine recall of the binomial expansion formula with minimal problem-solving, making it easier than average but not trivial since it requires correct setup and algebraic manipulation.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
1 In the expansion of \(( 2 + a x ) ^ { 7 }\), the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\). Find the value of the non-zero constant \(a\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(^7C_1 \times 2^6 \times a = ^7C_2 \times 2^5 \times a^2\)B2, 1, 0 Treat the same error in each expression as a single error
\(a = \left(\frac{7 \times 2^6}{21 \times 2^5}\right) = \frac{2}{3}\)B1 oe 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $^7C_1 \times 2^6 \times a = ^7C_2 \times 2^5 \times a^2$ | **B2, 1, 0** | Treat the same error in each expression as a single error |
| $a = \left(\frac{7 \times 2^6}{21 \times 2^5}\right) = \frac{2}{3}$ | **B1** | oe |

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1 In the expansion of $( 2 + a x ) ^ { 7 }$, the coefficient of $x$ is equal to the coefficient of $x ^ { 2 }$. Find the value of the non-zero constant $a$.

\hfill \mbox{\textit{CAIE P1 2014 Q1 [3]}}
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