CAIE P1 2017 November — Question 1 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2017
SessionNovember
Marks4
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Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeSingle binomial expansion
DifficultyModerate -0.3 This is a standard binomial expansion question requiring students to identify which term has x^0 by setting up the general term and solving for r. While it involves negative powers and requires careful algebraic manipulation, it's a routine textbook exercise with a clear method that AS-level students practice regularly.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
1 Find the term independent of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 9 }\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
EITHER: Term is \(^9C_3 \times 2^6 \times (-\frac{1}{4})^3\)(B1, B1, B1) OE
OR1: \(\left(\frac{8x^3-1}{4x^2}\right)^9 = \left(\frac{1}{4x^2}\right)^9(8x^3-1)^9\) or \(\left(\frac{1}{4x^2}\right)^9(1-8x^3)^9\)
Term is \(-\frac{1}{4^9} \times {}^9C_3 \times 8^6\)(B1, B1, B1) OE
OR2: \((2x)^9\left(1-\frac{1}{8x^3}\right)^9\)
Term is \(2^9 \times {}^9C_3 \times \left(-\frac{1}{8}\right)^3\)(B1, B1, B1) OE
Selected term, which must be independent of \(x\) = \(-84\)B1
4 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER:** Term is $^9C_3 \times 2^6 \times (-\frac{1}{4})^3$ | **(B1, B1, B1)** | OE |
| **OR1:** $\left(\frac{8x^3-1}{4x^2}\right)^9 = \left(\frac{1}{4x^2}\right)^9(8x^3-1)^9$ or $\left(\frac{1}{4x^2}\right)^9(1-8x^3)^9$ | | |
| Term is $-\frac{1}{4^9} \times {}^9C_3 \times 8^6$ | **(B1, B1, B1)** | OE |
| **OR2:** $(2x)^9\left(1-\frac{1}{8x^3}\right)^9$ | | |
| Term is $2^9 \times {}^9C_3 \times \left(-\frac{1}{8}\right)^3$ | **(B1, B1, B1)** | OE |
| Selected term, which must be independent of $x$ = $-84$ | **B1** | |
| | **4** | |

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1 Find the term independent of $x$ in the expansion of $\left( 2 x - \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 9 }$.\\

\hfill \mbox{\textit{CAIE P1 2017 Q1 [4]}}
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