CAIE P1 2023 June — Question 3 3 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2023
SessionJune
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCompleting the square and sketching
TypeDiscriminant for real roots condition
DifficultyModerate -0.8 This is a straightforward completing the square exercise with standard follow-up about discriminant/roots. Part (a) requires routine algebraic manipulation (factoring out 4, completing the square, expressing c in terms of p). Part (b) is immediate once (a) is done—just requires recognizing that no real roots means the completed square form has no solutions, giving p > 36. Both parts are textbook-standard with no problem-solving insight required, making this easier than average.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points
  1. Express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\). [2]
  2. Hence or otherwise find the set of values of \(p\) for which the equation \(4x^2 - 24x + p = 0\) has no real roots. [1]
Question 3:
AnswerMarks Guidance
3(a)4x32
seen or a4 and b3B1 OE Award marks for the correct expression or their values
 p 
a, b and c. Condone 4(x3) p36 = 0 and 4 9 .
 
 4 
AnswerMarks
–36 + p or p – 36 seen or c = p – 36B1
2
AnswerMarks Guidance
3(b)p360 leading to p36 or 242 44p 0 p 36 or 36 p B1
1
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
--- 3(a) ---
3(a) | 4x32
seen or a4 and b3 | B1 | OE Award marks for the correct expression or their values
 p 
a, b and c. Condone 4(x3) p36 = 0 and 4 9 .
 
 4 
–36 + p or p – 36 seen or c = p – 36 | B1
2
--- 3(b) ---
3(b) | p360 leading to p36 or 242 44p 0 p 36 or 36 p | B1 | Allow (36,) or 36 p. Consider final answer only.
1
Question | Answer | Marks | Guidance
\begin{enumerate}[label=(\alph*)]
\item Express $4x^2 - 24x + p$ in the form $a(x + b)^2 + c$, where $a$ and $b$ are integers and $c$ is to be given in terms of the constant $p$. [2]

\item Hence or otherwise find the set of values of $p$ for which the equation $4x^2 - 24x + p = 0$ has no real roots. [1]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2023 Q3 [3]}}
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