CAIE P1 2024 November — Question 3 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionNovember
Marks6
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TopicBinomial Theorem (positive integer n)
TypeProduct with unknown constant to determine
DifficultyModerate -0.8 This is a straightforward binomial expansion question requiring standard application of the binomial theorem formula. Part (a) involves direct calculation of two coefficients using nCr, while part (b) requires multiplying out and solving a simple equation. The question is routine with no conceptual challenges beyond basic binomial theorem mechanics, making it easier than average for A-level.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
  1. Find the coefficients of \(x^3\) and \(x^4\) in the expansion of \((3 - ax)^5\), where \(a\) is a constant. Give your answers in terms of \(a\). [3]
  2. Given that the coefficient of \(x^4\) in the expansion of \((ax + 7)(3 - ax)^5\) is 240, find the positive value of \(a\). [3]
Question 3:
AnswerMarks
3(a)5 5
x3:  32(−ax)3   −1032a3  or x4:  3(−ax)4   53a4 
AnswerMarks Guidance
3 4M1 Allow for either term, allow sign error and combination
notation.
AnswerMarks Guidance
x3: −90a3A1 Allow in the full expansion.
x4: 15a4A1 Allow in the full expansion.
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
3(b)Coefficient of x4 is atheir−90a3+7their15a4 =15a4
 M1 Must select two appropriate terms only.
15 a4 =240DM1 Reducing to a simple quartic equation in a.
 a4 =16  a=2A1 A0 if a=−2is given as a solution.
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
--- 3(a) ---
3(a) | 5 5
x3:  32(−ax)3   −1032a3  or x4:  3(−ax)4   53a4 
3 4 | M1 | Allow for either term, allow sign error and combination
notation.
x3: −90a3 | A1 | Allow in the full expansion.
x4: 15a4 | A1 | Allow in the full expansion.
3
Question | Answer | Marks | Guidance
--- 3(b) ---
3(b) | Coefficient of x4 is atheir−90a3+7their15a4 =15a4
  | M1 | Must select two appropriate terms only.
15 a4 =240 | DM1 | Reducing to a simple quartic equation in a.
 a4 =16  a=2 | A1 | A0 if a=−2is given as a solution.
3
Question | Answer | Marks | Guidance
\begin{enumerate}[label=(\alph*)]
\item Find the coefficients of $x^3$ and $x^4$ in the expansion of $(3 - ax)^5$, where $a$ is a constant. Give your answers in terms of $a$. [3]

\item Given that the coefficient of $x^4$ in the expansion of $(ax + 7)(3 - ax)^5$ is 240, find the positive value of $a$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q3 [6]}}
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