CAIE P1 2020 November — Question 5 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2020
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeCoefficients in arithmetic/geometric progression
DifficultyStandard +0.8 This question requires extracting binomial coefficients, setting up a geometric progression condition (middle term squared equals product of outer terms), and solving the resulting equation. It combines binomial theorem with GP properties in a non-routine way that requires algebraic manipulation beyond standard textbook exercises, though the individual components are A-level standard.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n1.04i Geometric sequences: nth term and finite series sum
5 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).
Question 5:
AnswerMarks Guidance
\([7C1a^6b(x)]\), \([7C2a^5b^2(x^2)]\), \([7C4a^3b^4(x^4)]\)B2, 1, 0 SOI, can be seen in an expansion
\(\frac{7C2a^5b^2(x^2)}{7C1a^6b(x)} = \frac{7C4a^3b^4(x^4)}{7C2a^5b^2(x^2)} \rightarrow \frac{21a^5b^2}{7a^6b} = \frac{35a^3b^4}{21a^5b^2}\)M1 A1 M1 for a correct relationship OE (Ft from *their* 3 terms). For A1 binomial coefficients must be correct & evaluated
\(\frac{a}{b} = \frac{5}{9}\)A1 OE 
## Question 5:

| $[7C1a^6b(x)]$, $[7C2a^5b^2(x^2)]$, $[7C4a^3b^4(x^4)]$ | B2, 1, 0 | SOI, can be seen in an expansion |
|---|---|---|
| $\frac{7C2a^5b^2(x^2)}{7C1a^6b(x)} = \frac{7C4a^3b^4(x^4)}{7C2a^5b^2(x^2)} \rightarrow \frac{21a^5b^2}{7a^6b} = \frac{35a^3b^4}{21a^5b^2}$ | M1 A1 | M1 for a correct relationship OE (Ft from *their* 3 terms). For A1 binomial coefficients must be correct & evaluated |
| $\frac{a}{b} = \frac{5}{9}$ | A1 | OE |

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5 In the expansion of $( a + b x ) ^ { 7 }$, where $a$ and $b$ are non-zero constants, the coefficients of $x , x ^ { 2 }$ and $x ^ { 4 }$ are the first, second and third terms respectively of a geometric progression.

Find the value of $\frac { a } { b }$.\\

\hfill \mbox{\textit{CAIE P1 2020 Q5 [5]}}
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