CAIE P3 2018 June — Question 7 7 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2018
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeComplete the square
DifficultyModerate -0.8 This is a straightforward multi-part question on completing the square and inverse functions. Part (i) is routine algebraic manipulation, parts (ii) and (iii) require simple recognition that the vertex gives the turning point and domain restriction, and part (iv) involves standard inverse function technique. All steps are textbook exercises with no problem-solving insight required, making it easier than average.
Spec1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence1.07n Stationary points: find maxima, minima using derivatives
The function f is defined by \(\mathrm{f} : x \mapsto 7 - 2x^2 - 12x\) for \(x \in \mathbb{R}\).
  1. Express \(7 - 2x^2 - 12x\) in the form \(a - 2(x + b)^2\), where \(a\) and \(b\) are constants. [2]
  2. State the coordinates of the stationary point on the curve \(y = \mathrm{f}(x)\). [1]
The function g is defined by \(\mathrm{g} : x \mapsto 7 - 2x^2 - 12x\) for \(x \geqslant k\).
  1. State the smallest value of \(k\) for which g has an inverse. [1]
  2. For this value of \(k\), find \(\mathrm{g}^{-1}(x)\). [3]
Question 7:
AnswerMarks Guidance
7(i)Substitute in uv, expand the product and use i2 = −1 M1
Obtain answer uv = −11−5 3iA1
EITHER: Substitute in u/v and multiply numerator and
denominator by the conjugate of v, or
AnswerMarks
equivalentM1
Obtain numerator −7+7 3i or denominator 7A1
Obtain final answer −1+ 3iA1
OR: Substitute in u/v , equate to x + iy and solve
AnswerMarks Guidance
for x or for yM1  −3 3= 3x−2y
 1=2x+ 3y
AnswerMarks
Obtain x = – 1 or y = 3A1
Obtain final answer −1+ 3iA1
5
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
7(ii)Show the points A and B representing u and v in relatively
correct positionsB1
Carry out a complete method for finding angle AOB, e.g.
calculate arg(u/v)
( )
( )
If using θ=tan−1 − 3 must refer to arg u
AnswerMarks Guidance
vM1 −1 2
−
tana= −1 ,tanb= 2 ⇒tan ( a−b )= 3 3 3
OR: 3 3 3 1− 2
9
=− 3
⇒θ= 2π
3
−3 3 3
  
OR: cosθ=   1     2   = −9+2 = −1
7 28 14 2
⇒θ= 2π
3
OR: cosθ= 28+7−49 =− 1 ⇒ θ= 2π
2 28 7 2 3
AnswerMarks Guidance
Prove the given statementA1 Given answer so check working carefully
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 7:
--- 7(i) ---
7(i) | Substitute in uv, expand the product and use i2 = −1 | M1
Obtain answer uv = −11−5 3i | A1
EITHER: Substitute in u/v and multiply numerator and
denominator by the conjugate of v, or
equivalent | M1
Obtain numerator −7+7 3i or denominator 7 | A1
Obtain final answer −1+ 3i | A1
OR: Substitute in u/v , equate to x + iy and solve
for x or for y | M1 |  −3 3= 3x−2y

 1=2x+ 3y
Obtain x = – 1 or y = 3 | A1
Obtain final answer −1+ 3i | A1
5
Question | Answer | Marks | Guidance
--- 7(ii) ---
7(ii) | Show the points A and B representing u and v in relatively
correct positions | B1
Carry out a complete method for finding angle AOB, e.g.
calculate arg(u/v)
( )
( )
If using θ=tan−1 − 3 must refer to arg u
v | M1 | −1 2
−
tana= −1 ,tanb= 2 ⇒tan ( a−b )= 3 3 3
OR: 3 3 3 1− 2
9
=− 3
⇒θ= 2π
3
−3 3 3
  
OR: cosθ=   1     2   = −9+2 = −1
7 28 14 2
⇒θ= 2π
3
OR: cosθ= 28+7−49 =− 1 ⇒ θ= 2π
2 28 7 2 3
Prove the given statement | A1 | Given answer so check working carefully
3
Question | Answer | Marks | Guidance
The function f is defined by $\mathrm{f} : x \mapsto 7 - 2x^2 - 12x$ for $x \in \mathbb{R}$.

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

\item State the coordinates of the stationary point on the curve $y = \mathrm{f}(x)$. [1]
\end{enumerate}

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

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item State the smallest value of $k$ for which g has an inverse. [1]

\item For this value of $k$, find $\mathrm{g}^{-1}(x)$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2018 Q7 [7]}}
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