AQA Paper 3 2020 June — Question 9 5 marks

Exam BoardAQA
ModulePaper 3 (Paper 3)
Year2020
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTrig Proofs
TypeProve trigonometric identity
DifficultyStandard +0.3 Part (a) is a standard trigonometric identity proof requiring double angle formulas and algebraic manipulation over 4 marksβ€”routine for A-level but requires careful working. Part (b) is a simple conceptual check about domain restrictions. Overall slightly easier than average as it follows well-established proof techniques without requiring novel insight.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae
  1. For \(\cos \theta \neq 0\), prove that $$\cosec 2\theta + \cot 2\theta = \cot \theta$$ [4 marks]
  2. Explain why $$\cot \theta \neq \cosec 2\theta + \cot 2\theta$$ when \(\cos \theta = 0\) [1 mark]
Question 9:
AnswerMarks
9(a)Uses cosec2ΞΈ and
1
cot2ΞΈ=
= sin2πœƒπœƒ
AnswerMarks Guidance
cos2πœƒπœƒ1.2 B1
1 cos2πœƒπœƒ
sin2πœƒπœƒ sin2πœƒπœƒ
=
1+ cos2πœƒπœƒ
sin2πœƒπœƒ
=
2 2
1+ 𝑐𝑐𝑐𝑐𝑐𝑐 πœƒπœƒβˆ’π‘π‘π‘ π‘ π‘™π‘™ πœƒπœƒ
2sinπœƒπœƒπ‘π‘π‘π‘π‘π‘πœƒπœƒ
=
2
2𝑐𝑐𝑐𝑐𝑐𝑐 πœƒπœƒ
2sinπœƒπœƒπ‘π‘π‘π‘π‘π‘πœƒπœƒ
= =
π‘π‘π‘π‘π‘π‘πœƒπœƒ
sinπœƒπœƒ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
sin2πœƒπœƒ
Uses the identity for
sin2ΞΈ = 2sinΞΈcosΞΈ or an identity
for cos2ΞΈ = cos2ΞΈ – sin2ΞΈ or
2cos2ΞΈ – 1 or 1 – 2sin2ΞΈ to
AnswerMarks Guidance
commence proof2.1 M1
Uses the identities for sin2ΞΈ and
AnswerMarks Guidance
cos2ΞΈ in correct proof1.1b A1
Completes a reasoned
argument leading to a single
trigonometric fraction to prove
given identity
AnswerMarks Guidance
AG2.1 R1
AnswerMarks
9(b)Deduces that when cosΞΈ=0
thencotΞΈis defined/zero/exists
on LHS but cosec2ΞΈ or cot2ΞΈ or
is undefined on
1 1
RHS
2sinΞΈcosΞΈ π‘π‘π‘Žπ‘Ž sin2ΞΈ
or
deduces that LHS is defined but
RHS is undefined
Must compare both LHS and
AnswerMarks Guidance
RHS2.2a E1
cotΞΈ= 0 on LHS but because the
value of sin2ΞΈ =0, cosec2ΞΈ and
cot2ΞΈ are undefined on RHS.
AnswerMarks Guidance
Total5
QMarking Instructions AO 
Question 9:
--- 9(a) ---
9(a) | Uses cosec2ΞΈ and
1
cot2ΞΈ=
= sin2πœƒπœƒ
cos2πœƒπœƒ | 1.2 | B1 | +
1 cos2πœƒπœƒ
sin2πœƒπœƒ sin2πœƒπœƒ
=
1+ cos2πœƒπœƒ
sin2πœƒπœƒ
=
2 2
1+ 𝑐𝑐𝑐𝑐𝑐𝑐 πœƒπœƒβˆ’π‘π‘π‘ π‘ π‘™π‘™ πœƒπœƒ
2sinπœƒπœƒπ‘π‘π‘π‘π‘π‘πœƒπœƒ
=
2
2𝑐𝑐𝑐𝑐𝑐𝑐 πœƒπœƒ
2sinπœƒπœƒπ‘π‘π‘π‘π‘π‘πœƒπœƒ
= =
π‘π‘π‘π‘π‘π‘πœƒπœƒ
sinπœƒπœƒ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
sin2πœƒπœƒ
Uses the identity for
sin2ΞΈ = 2sinΞΈcosΞΈ or an identity
for cos2ΞΈ = cos2ΞΈ – sin2ΞΈ or
2cos2ΞΈ – 1 or 1 – 2sin2ΞΈ to
commence proof | 2.1 | M1
Uses the identities for sin2ΞΈ and
cos2ΞΈ in correct proof | 1.1b | A1
Completes a reasoned
argument leading to a single
trigonometric fraction to prove
given identity
AG | 2.1 | R1
--- 9(b) ---
9(b) | Deduces that when cosΞΈ=0
thencotΞΈis defined/zero/exists
on LHS but cosec2ΞΈ or cot2ΞΈ or
is undefined on
1 1
RHS
2sinΞΈcosΞΈ π‘π‘π‘Žπ‘Ž sin2ΞΈ
or
deduces that LHS is defined but
RHS is undefined
Must compare both LHS and
RHS | 2.2a | E1 | When cosΞΈ=0the value of
cotΞΈ= 0 on LHS but because the
value of sin2ΞΈ =0, cosec2ΞΈ and
cot2ΞΈ are undefined on RHS.
Total | 5
Q | Marking Instructions | AO | Marks | Typical Solution
\begin{enumerate}[label=(\alph*)]
\item For $\cos \theta \neq 0$, prove that
$$\cosec 2\theta + \cot 2\theta = \cot \theta$$
[4 marks]

\item Explain why
$$\cot \theta \neq \cosec 2\theta + \cot 2\theta$$
when $\cos \theta = 0$
[1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2020 Q9 [5]}}
This paper (18 questions)
View full paper
Q1 1 Q2 1 Q3 1 Q4 7 Q5 9 Q6 7 Q7 7 Q8 12 Q9 5 Q10 1 Q11 1 Q12 4 Q13 6 Q14 7 Q15 5 Q16 4 Q17 8 Q18 14