CAIE P1 2018 November — Question 9 7 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2018
SessionNovember
Marks7
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Mark schemeDownload PDF ↗
TopicCompleting the square and sketching
TypeComplete square then find vertex/turning point
DifficultyModerate -0.8 This is a straightforward completing the square question with standard follow-ups about range and inverse functions. All parts use routine techniques taught in P1: algebraic manipulation for completing the square, reading the range from completed square form, identifying the vertex for the domain restriction, and finding an inverse by swapping and rearranging. No problem-solving insight required, just methodical application of standard procedures.
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 existence
The function f is defined by \(\text{f} : x \mapsto 2x^2 - 12x + 7\) for \(x \in \mathbb{R}\).
  1. Express \(2x^2 - 12x + 7\) in the form \(2(x + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
  2. State the range of f. [1]
The function g is defined by \(\text{g} : x \mapsto 2x^2 - 12x + 7\) for \(x \leqslant k\).
  1. State the largest value of \(k\) for which g has an inverse. [1]
  2. Given that g has an inverse, find an expression for \(\text{g}^{-1}(x)\). [3]
Question 9:
AnswerMarks Guidance
9(i)( )2
2x² − 12x + 7 = 2 x−3 −11B1 B1 Mark full expression if present: B1 for 2(x – 3)2 and B1
for – 11.
If no clear expression award a = – 3 and b = – 11.
2
AnswerMarks Guidance
9(ii)Range (of f or y) ⩾ ‘their – 11’ B1FT
Do NOT accept x > or ⩾
1
AnswerMarks Guidance
9(iii)(k =) –“their a” also allow x or k ⩽ 3 B1FT
FT for their “a” or start again using =0.
dx
Do NOT accept x = 3.
1
AnswerMarks
9(iv)( )2
y = 2 x−3 −11 → y + 11 = 2(x – 3)²
y+11
=( )
x−3 ²
AnswerMarks Guidance
2*M1 Isolating their (x – 3)², condone – 11.
y+11 y+11
 
x = 3+   or 3−  
AnswerMarks Guidance
 2   2 DM1 Other operations in correct order, allow ± at this stage.
Condone – 3.
x+11
(g−1(x) or y) = 3−  
AnswerMarks Guidance
 2 A1 needs ‘–’. x and y could be interchanged at the start.
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 9:
--- 9(i) ---
9(i) | ( )2
2x² − 12x + 7 = 2 x−3 −11 | B1 B1 | Mark full expression if present: B1 for 2(x – 3)2 and B1
for – 11.
If no clear expression award a = – 3 and b = – 11.
2
--- 9(ii) ---
9(ii) | Range (of f or y) ⩾ ‘their – 11’ | B1FT | FT for their ‘b’ or start again. Condone >.
Do NOT accept x > or ⩾
1
--- 9(iii) ---
9(iii) | (k =) –“their a” also allow x or k ⩽ 3 | B1FT | dy
FT for their “a” or start again using =0.
dx
Do NOT accept x = 3.
1
--- 9(iv) ---
9(iv) | ( )2
y = 2 x−3 −11 → y + 11 = 2(x – 3)²
y+11
=( )
x−3 ²
2 | *M1 | Isolating their (x – 3)², condone – 11.
y+11 y+11
 
x = 3+   or 3−  
 2   2  | DM1 | Other operations in correct order, allow ± at this stage.
Condone – 3.
x+11
(g−1(x) or y) = 3−  
 2  | A1 | needs ‘–’. x and y could be interchanged at the start.
3
Question | Answer | Marks | Guidance
The function f is defined by $\text{f} : x \mapsto 2x^2 - 12x + 7$ for $x \in \mathbb{R}$.

\begin{enumerate}[label=(\roman*)]
\item Express $2x^2 - 12x + 7$ in the form $2(x + a)^2 + b$, where $a$ and $b$ are constants. [2]
\item State the range of f. [1]
\end{enumerate}

The function g is defined by $\text{g} : x \mapsto 2x^2 - 12x + 7$ for $x \leqslant k$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumii}{2}
\item State the largest value of $k$ for which g has an inverse. [1]
\item Given that g has an inverse, find an expression for $\text{g}^{-1}(x)$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P1 2018 Q9 [7]}}
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