CAIE P1 Specimen — Question 1 3 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
SessionSpecimen
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeCoefficient zero by design (proof)
DifficultyStandard +0.8 This requires expanding two binomial expressions, collecting terms systematically, and proving a specific coefficient vanishes through algebraic manipulation. It demands careful organization and algebraic insight beyond routine expansion, but uses only standard binomial theorem techniques.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n

1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.

Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\((a+x)^5 = a^5 + {}^5C_1a^4x + {}^5C_2a^3x^2 + \ldots\) soiM1 Ignore subsequent terms
\(\left(-\frac{2}{a} \times (\text{their } 5a^4) + (\text{their } 10a^3)\right)(x^2)\)M1
\(0\)A1 AG
Total: 3
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(a+x)^5 = a^5 + {}^5C_1a^4x + {}^5C_2a^3x^2 + \ldots$ soi | M1 | Ignore subsequent terms |
| $\left(-\frac{2}{a} \times (\text{their } 5a^4) + (\text{their } 10a^3)\right)(x^2)$ | M1 | |
| $0$ | A1 | AG |
| **Total: 3** | | |

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1 In the expansion of $\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }$, where $a$ is a non-zero constant, show that the coefficient of $x ^ { 2 }$ is zero.\\

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