Edexcel P2 2023 June — Question 4 7 marks

Exam BoardEdexcel
ModuleP2 (Pure Mathematics 2)
Year2023
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicBinomial Theorem (positive integer n)
TypeFind constants from coefficient conditions on terms
DifficultyModerate -0.3 This is a straightforward binomial expansion question requiring students to find coefficients using the binomial theorem formula, then solve a simple equation relating two coefficients. The algebra is routine and the problem-solving demand is minimal—students just need to apply the standard formula and solve B=18D. Slightly easier than average due to its mechanical nature, though the constraint B=18D adds a small problem-solving element beyond pure recall.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
  1. The binomial expansion, in ascending powers of \(x\), of
$$( 3 + p x ) ^ { 5 }$$ where \(p\) is a constant, can be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.
  1. Find the value of \(A\) Given that
    • \(B = 18 D\)
    • \(p < 0\)
    • find
      1. the value of \(p\)
      2. the value of \(C\)
Question 4:
Part (a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(3^5\) or \(243\)B1 Do not award if outside the bracket as \(3^5\left(1+\frac{px}{3}\right)^5\); allow if some expansion terms found and 243 is constant term
Part (b)(i):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((B=)5\times3^4p\ (=405p)\) or \((D=)10\times3^2p^3\ (=90p^3)\)M1A1 Simplified or unsimplified including unprocessed indices; binomial notation \({}^5C_1\) or \({}^5C_3\) must be evaluated
\(B=18D \Rightarrow 405p=18\times90p^3 \Rightarrow p^2=\frac{1}{4} \Rightarrow p=-\frac{1}{2}\)M1A1 Uses \("B"=18"D"\) to form cubic in \(p\); divides by \(p\); \(p=-\frac{1}{2}\) only, positive root and \(p=0\) rejected
Part (b)(ii):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((C=){}^5C_2\times3^3\times\left(-\frac{1}{2}\right)^2=\frac{135}{2}\)M1A1 Uses \(p^2\), \(p\) or \( 
# Question 4:

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3^5$ or $243$ | B1 | Do not award if outside the bracket as $3^5\left(1+\frac{px}{3}\right)^5$; allow if some expansion terms found and 243 is constant term |

## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(B=)5\times3^4p\ (=405p)$ or $(D=)10\times3^2p^3\ (=90p^3)$ | M1A1 | Simplified or unsimplified including unprocessed indices; binomial notation ${}^5C_1$ or ${}^5C_3$ must be evaluated |
| $B=18D \Rightarrow 405p=18\times90p^3 \Rightarrow p^2=\frac{1}{4} \Rightarrow p=-\frac{1}{2}$ | M1A1 | Uses $"B"=18"D"$ to form cubic in $p$; divides by $p$; $p=-\frac{1}{2}$ only, positive root and $p=0$ rejected |

## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(C=){}^5C_2\times3^3\times\left(-\frac{1}{2}\right)^2=\frac{135}{2}$ | M1A1 | Uses $p^2$, $p$ or $|p|$; attempts ${}^5C_2\times3^3p^2$; answer $\frac{135}{2}$ or $67.5$ or $67\frac{1}{2}$ |

---
\begin{enumerate}
  \item The binomial expansion, in ascending powers of $x$, of
\end{enumerate}

$$( 3 + p x ) ^ { 5 }$$

where $p$ is a constant, can be written in the form

$$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$

where $A$, $B$, $C$ and $D$ are constants.\\
(a) Find the value of $A$

Given that

\begin{itemize}
  \item $B = 18 D$
  \item $p < 0$\\
(b) find\\
(i) the value of $p$\\
(ii) the value of $C$
\end{itemize}

\hfill \mbox{\textit{Edexcel P2 2023 Q4 [7]}}
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