AQA Paper 2 2018 June — Question 8 10 marks

Exam BoardAQA
ModulePaper 2 (Paper 2)
Year2018
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicHarmonic Form
TypeRange of rational function with harmonic denominator
DifficultyStandard +0.8 This question requires expressing a linear combination of sin and cos in the form R sin(x + α) + k, then applying transformations and finding reciprocal extrema. While the technique is standard A-level content, the √3 coefficient creates non-trivial arithmetic, the 7-mark transformation justification demands careful reasoning about order and composition, and finding reciprocal extrema requires understanding that min/max swap. This is more demanding than typical transformation questions but doesn't require novel insight beyond curriculum content.
Spec1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc
  1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt{3} \sin x - 3 \cos x + 4\) Fully justify your answer. [7 marks]
    1. Show that the least value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) is \(\frac{2 - \sqrt{3}}{2}\) [2 marks]
    2. Find the greatest value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) [1 mark]
Question 8:
AnswerMarks Guidance
8(a)Compares with Rcos(x) or
Rsin(x)AO3.1a M1
RsinxcosRcosxsin
Rcos 3
Rsin3
R 122 3
tan 3

3
y2 3sin(x )4
3

 
Translation 3
 
 
 0
Stretch in the y-direction scale
factor 2 3
0
Translation  
4
Obtains two correct equations for
Randfor example
Rcos 3
Rsin3
AnswerMarks Guidance
Must be explicitly seenAO3.1a M1
Obtains correct R
Condone AWRT 3.46 PI by
AnswerMarks Guidance
description of stretchAO1.1b B1
Obtains correct in radians or
degrees PI by description of
AnswerMarks Guidance
translationAO1.1b B1
Interprets their values of Rand
to form an equation of the form
y  Rsin(x)4or
AnswerMarks Guidance
y  Rcos(x)4AO3.2a B1F
Interprets ‘their’ equation to identify
AnswerMarks Guidance
a transformationAO3.2a E1F
Identifies all required
transformations in a correct order
AnswerMarks Guidance
CAOAO3.2a A1
(b)(i)Deduces the least value occurs
when their sin(x)1
3
Using ‘their’ values of R and α
1
PI by sight of
AnswerMarks Guidance
2 34AO2.2a M1
3sinx3cosx4 2 3sin(x)4
3
Least value when sin(x)1
3
least value is given by
1 2 3
2 34 2
Completes rigorous argument to
1
obtain and then the given
2 34
AnswerMarks Guidance
answerAO2.1 R1
(b)(ii)Deduces the greatest value
Using ‘their’ values of R and α
1 2 3
ACF 
AnswerMarks Guidance
2 34 2AO2.2a B1F
Greatest value =
2
AnswerMarks Guidance
Total10
QMarking instructions AO 
Question 8:
--- 8(a) ---
8(a) | Compares with Rcos(x) or
Rsin(x) | AO3.1a | M1 | 3sinx3cosxRsin(x)
RsinxcosRcosxsin
Rcos 3
Rsin3
R 122 3
tan 3


3

y2 3sin(x )4
3

 
Translation 3
 
 
 0
Stretch in the y-direction scale
factor 2 3
0
Translation  
4
Obtains two correct equations for
Randfor example
Rcos 3
Rsin3
Must be explicitly seen | AO3.1a | M1
Obtains correct R
Condone AWRT 3.46 PI by
description of stretch | AO1.1b | B1
Obtains correct in radians or
degrees PI by description of
translation | AO1.1b | B1
Interprets their values of Rand
to form an equation of the form
y  Rsin(x)4or
y  Rcos(x)4 | AO3.2a | B1F
Interprets ‘their’ equation to identify
a transformation | AO3.2a | E1F
Identifies all required
transformations in a correct order
CAO | AO3.2a | A1
(b)(i) | Deduces the least value occurs
when their sin(x)1
3
Using ‘their’ values of R and α
1
PI by sight of
2 34 | AO2.2a | M1 | 1 1

3sinx3cosx4 2 3sin(x)4
3
Least value when sin(x)1
3
least value is given by
1 2 3

2 34 2
Completes rigorous argument to
1
obtain and then the given
2 34
answer | AO2.1 | R1
(b)(ii) | Deduces the greatest value
Using ‘their’ values of R and α
1 2 3
ACF 
2 34 2 | AO2.2a | B1F | 2 3
Greatest value =
2
Total | 10
Q | Marking instructions | AO | Mark | Typical solution
\begin{enumerate}[label=(\alph*)]
\item Determine a sequence of transformations which maps the graph of $y = \sin x$ onto the graph of $y = \sqrt{3} \sin x - 3 \cos x + 4$

Fully justify your answer.
[7 marks]

\item \begin{enumerate}[label=(\roman*)]
\item Show that the least value of $\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}$ is $\frac{2 - \sqrt{3}}{2}$
[2 marks]

\item Find the greatest value of $\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}$
[1 mark]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 2 2018 Q8 [10]}}
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