CAIE P1 2024 November — Question 11 11 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionNovember
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeComplete the square
DifficultyModerate -0.3 This is a standard multi-part question on quadratic functions covering completing the square, transformations, domain restrictions for inverses, and finding inverse functions. All techniques are routine for P1 level with no novel problem-solving required, though the multi-step nature and 11 total marks place it slightly below average difficulty.
Spec1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)
The function f is defined by f\((x) = 3 + 6x - 2x^2\) for \(x \in \mathbb{R}\).
  1. Express f\((x)\) in the form \(a - b(x - c)^2\), where \(a\), \(b\) and \(c\) are constants, and state the range of f. [3]
  2. The graph of \(y = \)f\((x)\) is transformed to the graph of \(y = \)h\((x)\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \)h\((x)\) has a minimum point at the origin. Give details of the reflection and translation involved. [2] The function g is defined by g\((x) = 3 + 6x - 2x^2\) for \(x \leqslant 0\).
  3. Sketch the graph of \(y = \)g\((x)\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]
  4. Sketch the graph of \(y = \)g\(^{-1}(x)\) on your diagram in (c), and find an expression for g\(^{-1}(x)\). You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]
Question 11:
AnswerMarks
11(a)3
Obtain b=2 and c=
AnswerMarks
2B1
2
15  3
Obtain −2x− 
AnswerMarks
2  2B1
15 15
State range is y or f (x) with ⩽ given or clearly implied (not <)
AnswerMarks Guidance
2 2B1 FT Following their value of a.
3
AnswerMarks Guidance
11(b)State that reflection is in x-axis B1
 3
−
 
2
State or imply that translation is by   or equivalent
15 
 
AnswerMarks Guidance
 2 B1 FT Following their values of a and c in part (a).
Accept transformations in any order.
2
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
11(c)Sketch the correct graph appearing in second and third quadrants only B1
State that each y-value is associated with a single x-value or equivalentB1 Accept passes the horizontal line test.
Ignore passes the vertical line test.
2
AnswerMarks Guidance
11(d)Sketch the correct graph with suitable labelling to distinguish the two curves B1
Draw the line y=xB1 See above; no need to label the line.
Attempt correct process for finding the inverse functionM1 Allowing use of  and y so far.
3 15 1
Obtain − − x or equivalent
AnswerMarks Guidance
2 4 2A1 Must involve x at the conclusion.
4
Question 11:
--- 11(a) ---
11(a) | 3
Obtain b=2 and c=
2 | B1
2
15  3
Obtain −2x− 
2  2 | B1
15 15
State range is y or f (x) with ⩽ given or clearly implied (not <)
2 2 | B1 FT | Following their value of a.
3
--- 11(b) ---
11(b) | State that reflection is in x-axis | B1 | Accept transformations in any order.
 3
−
 
2
State or imply that translation is by   or equivalent
15 
 
 2  | B1 FT | Following their values of a and c in part (a).
Accept transformations in any order.
2
Question | Answer | Marks | Guidance
--- 11(c) ---
11(c) | Sketch the correct graph appearing in second and third quadrants only | B1
State that each y-value is associated with a single x-value or equivalent | B1 | Accept passes the horizontal line test.
Ignore passes the vertical line test.
2
--- 11(d) ---
11(d) | Sketch the correct graph with suitable labelling to distinguish the two curves | B1 | Appearing in third and fourth quadrants only.
Draw the line y=x | B1 | See above; no need to label the line.
Attempt correct process for finding the inverse function | M1 | Allowing use of  and y so far.
3 15 1
Obtain − − x or equivalent
2 4 2 | A1 | Must involve x at the conclusion.
4
The function f is defined by f$(x) = 3 + 6x - 2x^2$ for $x \in \mathbb{R}$.

\begin{enumerate}[label=(\alph*)]
\item Express f$(x)$ in the form $a - b(x - c)^2$, where $a$, $b$ and $c$ are constants, and state the range of f. [3]

\item The graph of $y = $f$(x)$ is transformed to the graph of $y = $h$(x)$ by a reflection in one of the axes followed by a translation. It is given that the graph of $y = $h$(x)$ has a minimum point at the origin.

Give details of the reflection and translation involved. [2]

The function g is defined by g$(x) = 3 + 6x - 2x^2$ for $x \leqslant 0$.

\item Sketch the graph of $y = $g$(x)$ and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]

\item Sketch the graph of $y = $g$^{-1}(x)$ on your diagram in (c), and find an expression for g$^{-1}(x)$. You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2024 Q11 [11]}}
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