Question 1 - A-Level Maths

OCR MEI Paper 2 2022 June — Question 1 4 marks

Exam Board	OCR MEI
Module	Paper 2 (Paper 2)
Year	2022
Session	June
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Harmonic Form
Type	Express in harmonic form
Difficulty	Moderate -0.8 This is a standard R-formula question requiring straightforward application of the compound angle formula. Students expand R cos(θ - α), equate coefficients to find R = 2 and α = π/3, with no problem-solving or insight needed beyond direct method application.
Spec	1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

Express \(\cos\theta + \sqrt{3}\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R\) and \(\alpha\) are exact values to be determined. [4]

Show mark scheme Show mark scheme source

Question 1:

Answer	Marks
1	2

𝑅2 = 12 +√3

√3 √3 1

tan𝛼 = or sin𝛼 = or cos𝛼 = soi

1 2 2

𝜋

R = 2 or 𝛼 = or 𝛼 = 60 ֠ seen

3 𝜋

2cos(θ ‒ ) or 2cos(θ ‒ 60 ֠ ) isw

Answer	Marks
3	M1

M1

A1

Answer	Marks
A1	1.1

1.1

Answer	Marks
1.1	may be implied by correct answer

may see eg 𝛼 = tan−1( √3 )

1 may be implied by correct answer

[4]

Answer	Marks	Guidance
Alternatively	M1	for equating coefficients

cos𝜃 +√3sin𝜃 = 𝑅cos𝜃cos𝛼 +𝑅sin𝜃sin𝛼

so 1 = 𝑅cos𝛼 and √3 = 𝑅sin𝛼

1 √3

=

Answer	Marks	Guidance
cos𝛼 sin𝛼	M1	for eliminating R

𝜋

𝛼 = or 𝛼 = 60 ֠ seen

Answer	Marks
3	A1

𝜋

2cos(θ ‒ ) or 2cos(θ ‒ 60 ֠ ) isw

Answer	Marks
3	A1

Question 1:
1 | 2
𝑅2 = 12 +√3
√3 √3 1
tan𝛼 = or sin𝛼 = or cos𝛼 = soi
1 2 2
𝜋
R = 2 or 𝛼 = or 𝛼 = 60 ֠ seen
3
𝜋
2cos(θ ‒ ) or 2cos(θ ‒ 60 ֠ ) isw
3 | M1
M1
A1
A1 | 1.1
1.1
1.1
1.1 | may be implied by correct answer
may see eg 𝛼 = tan−1( √3 )
1
may be implied by correct answer
[4]
Alternatively | M1 | for equating coefficients
cos𝜃 +√3sin𝜃 = 𝑅cos𝜃cos𝛼 +𝑅sin𝜃sin𝛼
so 1 = 𝑅cos𝛼 and √3 = 𝑅sin𝛼
1 √3
=
cos𝛼 sin𝛼 | M1 | for eliminating R
𝜋
𝛼 = or 𝛼 = 60 ֠ seen
3 | A1
𝜋
2cos(θ ‒ ) or 2cos(θ ‒ 60 ֠ ) isw
3 | A1

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Express $\cos\theta + \sqrt{3}\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R$ and $\alpha$ are exact values to be determined. [4]

\hfill \mbox{\textit{OCR MEI Paper 2 2022 Q1 [4]}}

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Q1 4 Q2 2 Q3 6 Q4 4 Q5 3 Q6 2 Q7 2 Q8 3 Q9 9 Q10 7 Q11 10 Q12 8 Q13 8 Q14 8 Q15 9 Q16 15