AQA Paper 2 Specimen — Question 5 9 marks

Exam BoardAQA
ModulePaper 2 (Paper 2)
SessionSpecimen
Marks9
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Mark schemeDownload PDF ↗
TopicHarmonic Form
TypeRange of squared harmonic expression
DifficultyStandard +0.3 Part (a) requires expressing 3cos θ + 3sin θ in the form R cos(θ - α), which is a standard A-level technique, then identifying the transformations (stretch and translation). Part (b) applies this result to find max/min values using straightforward substitution. While it requires multiple steps and clear justification, both parts follow well-established procedures taught explicitly in the curriculum with no novel problem-solving required.
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 \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
Question 5:
AnswerMarks Guidance
5(a)Compares with Rcos() or
Rsin()AO3.1a M1
 RsincosRsinsin
Rcos3 and Rsin3
R 18

4
 
3cos3sin 18cos  
 4
Which is a stretch in the y-direction
scale factor 18

 
and a translation 4
 
0
Identifies version which will
AnswerMarks Guidance
allow them to solve the problemAO3.1a A1
Obtains correct RAO1.1b A1
Obtains correct AO1.1b A1
Interprets ‘their’ equation to
AnswerMarks Guidance
identify first transformationAO3.2a E1
Interprets ‘their’ equation to
AnswerMarks Guidance
identify second transformationAO3.2a E1
(b)Constructs a rigorous
mathematical argument, to find
either the least or greatest value
Only award if they have a
completely correct solution,
which is clear, easy to follow
and contains no slips (no FT for
AnswerMarks Guidance
this mark)AO2.1 R1
2
  
4  18cos  
  4
Least value occurs when
 
cos2  0
 
 4
∴least value = 4
Greatest value occurs when
 
cos2  1
 
 4
greatest value = 4 + 18
= 22
Deduces the least value
AnswerMarks Guidance
Using ‘their’ values of R and AO2.2a A1F
Deduces the greatest value
AnswerMarks Guidance
Using ‘their’ values of R and AO2.2a A1F
Total9
QMarking Instructions AO 
Question 5:
--- 5(a) ---
5(a) | Compares with Rcos() or
Rsin() | AO3.1a | M1 | Rcos()
 RsincosRsinsin
Rcos3 and Rsin3
R 18


4
 
3cos3sin 18cos  
 4
Which is a stretch in the y-direction
scale factor 18

 
and a translation 4
 
0
Identifies version which will
allow them to solve the problem | AO3.1a | A1
Obtains correct R | AO1.1b | A1
Obtains correct  | AO1.1b | A1
Interprets ‘their’ equation to
identify first transformation | AO3.2a | E1
Interprets ‘their’ equation to
identify second transformation | AO3.2a | E1
(b) | Constructs a rigorous
mathematical argument, to find
either the least or greatest value
Only award if they have a
completely correct solution,
which is clear, easy to follow
and contains no slips (no FT for
this mark) | AO2.1 | R1 | 4(3cos3sin2
2
  
4  18cos  
  4
Least value occurs when
 
cos2  0
 
 4
∴least value = 4
Greatest value occurs when
 
cos2  1
 
 4
greatest value = 4 + 18
= 22
Deduces the least value
Using ‘their’ values of R and  | AO2.2a | A1F
Deduces the greatest value
Using ‘their’ values of R and  | AO2.2a | A1F
Total | 9
Q | Marking Instructions | AO | Marks | Typical Solution
\begin{enumerate}[label=(\alph*)]
\item Determine a sequence of transformations which maps the graph of $y = \cos \theta$ onto the graph of $y = 3\cos \theta + 3\sin \theta$

Fully justify your answer.
[6 marks]

\item Hence or otherwise find the least value and greatest value of
$$4 + (3\cos \theta + 3\sin \theta)^2$$

Fully justify your answer.
[3 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 2  Q5 [9]}}
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