CAIE P1 2024 June — Question 9 8 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTangents, normals and gradients
TypeIncreasing/decreasing intervals
DifficultyModerate -0.3 Part (a) requires solving a quadratic inequality after expanding and simplifying f'(x) < 0, which is straightforward A-level technique. Part (b) involves integrating a polynomial (after expansion) and using an initial condition - both standard procedures. The algebra is slightly involved but this is a typical textbook-style question testing core differentiation/integration skills without requiring problem-solving insight.
Spec1.07o Increasing/decreasing: functions using sign of dy/dx1.08a Fundamental theorem of calculus: integration as reverse of differentiation
A function f is such that \(f'(x) = 6(2x-3)^2 - 6x\) for \(x \in \mathbb{R}\).
  1. Determine the set of values of \(x\) for which f\((x)\) is decreasing. [4]
  2. Given that f\((1) = -1\), find f\((x)\). [4]
Question 9:
AnswerMarks Guidance
9(a)62x32
6x0 or = 0B1* 62x32 62x32
Condone ⩽ 0. If only or 6 is
used, do not treat as a MR.
24x2 78x54 or 4x2 13x9 or x14x9
OR
AnswerMarks Guidance
62x32 6x leading to 2x3 x leading to 2x x3M1 Expanding brackets and collecting terms to arrive at
a three term quadratic, only condone sign errors.
9
x1 ,
AnswerMarks
4B1
9 9  9
1 x or x1 and x or  1, 
AnswerMarks Guidance
4 4  4DB1FT OE
Condone consistent use of ⩽ and ⩾ or [ ].
9 9
Do not allow x1 or x nor x1, x .
4 4
FT on their values coming from a correct initial
statement.
4
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
9(b) 6 2x33  6 
  fx      x2  C
AnswerMarks Guidance
32  2 B1 B1 B1 for each Correct integral.
113 312 CM1 fx1 equated to their integrated expression,
defined by two terms with at least one correct power
+ C, with x = 1.
f x2x33 3x2 3
AnswerMarks Guidance
 A1 CAO
Only condone C = 3 as final answer if coefficients
have been simplified earlier.
Do not ISW if the result is of the form ymxc.
Alternative method for Question 9(b)
f x 24x2 78x54 leading to fx8x3 39x2 54xC
AnswerMarks Guidance
   (B2,1,0) B2 completely correct, B1 any two correct terms.
183954C(M1) fx1 equated to their integrated expression,
defined by three terms with at least one correct
power + C, with x = 1.
f x 8x339x2 54x24
AnswerMarks Guidance
 (A1) Only condone C = 24 as final answer if
coefficients have been simplified earlier.
Do not ISW if the result is of the form ymxc.
4
AnswerMarks Guidance
QuestionAnswer Marks 
Question 9:
--- 9(a) ---
9(a) | 62x32
6x0 or = 0 | B1* | 62x32 62x32
Condone ⩽ 0. If only or 6 is
used, do not treat as a MR.
24x2 78x54 or 4x2 13x9 or x14x9
OR
62x32 6x leading to 2x3 x leading to 2x x3 | M1 | Expanding brackets and collecting terms to arrive at
a three term quadratic, only condone sign errors.
9
x1 ,
4 | B1
9 9  9
1 x or x1 and x or  1, 
4 4  4 | DB1FT | OE
Condone consistent use of ⩽ and ⩾ or [ ].
9 9
Do not allow x1 or x nor x1, x .
4 4
FT on their values coming from a correct initial
statement.
4
Question | Answer | Marks | Guidance
--- 9(b) ---
9(b) |  6 2x33  6 
  fx      x2  C
32  2  | B1 B1 | B1 for each Correct integral.
113 312 C | M1 | fx1 equated to their integrated expression,
defined by two terms with at least one correct power
+ C, with x = 1.
f x2x33 3x2 3
  | A1 | CAO
Only condone C = 3 as final answer if coefficients
have been simplified earlier.
Do not ISW if the result is of the form ymxc.
Alternative method for Question 9(b)
f x 24x2 78x54 leading to fx8x3 39x2 54xC
    | (B2,1,0) | B2 completely correct, B1 any two correct terms.
183954C | (M1) | fx1 equated to their integrated expression,
defined by three terms with at least one correct
power + C, with x = 1.
f x 8x339x2 54x24
  | (A1) | Only condone C = 24 as final answer if
coefficients have been simplified earlier.
Do not ISW if the result is of the form ymxc.
4
Question | Answer | Marks | Guidance
A function f is such that $f'(x) = 6(2x-3)^2 - 6x$ for $x \in \mathbb{R}$.

\begin{enumerate}[label=(\alph*)]
\item Determine the set of values of $x$ for which f$(x)$ is decreasing. [4]

\item Given that f$(1) = -1$, find f$(x)$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q9 [8]}}
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