Question 15 - A-Level Maths

AQA Paper 1 2024 June — Question 15 6 marks

Exam Board	AQA
Module	Paper 1 (Paper 1)
Year	2024
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Trig Proofs
Type	Complete or critique given proof
Difficulty	Standard +0.3 Part (a) is a routine trigonometric identity proof using standard double angle formulas and reciprocal identities (4 marks for straightforward algebraic manipulation). Part (b) tests understanding of domain restrictions: students must recognize that cos θ = 1 makes sec θ undefined in the original equation, then reject θ = 0° and 360° from the solution set. While this requires careful thinking about validity of solutions, it's a standard A-level skill with minimal computational demand.
Spec	1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 1.05l Double angle formulae: and compound angle formulae 1.05o Trigonometric equations: solve in given intervals

Show that the expression $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta$$ can be written as $$4 \cos \theta - \sec \theta$$ where $\sin \theta \neq 0$ and $\cos \theta \neq 0$ [4 marks]
A student is attempting to solve the equation $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$ They use the result from part (a), and write the following incorrect solution: $\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3$ Step 1: $4 \cos \theta - \sec \theta = 3$ Step 2: $4 \cos \theta - \frac{1}{\cos \theta} - 3 = 0$ Step 3: $4 \cos^2 \theta - 3 \cos \theta - 1 = 0$ Step 4: $\cos \theta = 1$ or $\cos \theta = -0.25$ Step 5: $\theta = 0°, 104.5°, 255.5°, 360°$
1. Explain why the student should reject one of their values for $\cos \theta$ in Step 4. [1 mark]
2. State the correct solutions to the equation $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$ [1 mark]

Show mark scheme Show mark scheme source

Question 15:

Answer	Marks
15(a)	Recalls the identity for

sin2=2sincosor an

identity for c o s  2 eg

cos2=2cos2−1

cos2=1−2sin2

c o s c o s s i n    2 = 2 − 2

This mark could be scored later

if compound angle formula is

used with a completely correct

Answer	Marks	Guidance
argument.	1.2	B1

=2sincoscosec+ ( 2cos2−1 ) sec

=2sincoscosec+2cos2sec−sec

=2cos+2cos−sec

=4cos−sec

Substitutes A s i n c o s   and a

correct identity for cos2

OR

Substitutes s i n c o s   2 and an

identity for cos2with sign

errors condoned provided

cos2is not replaced with an

expression equivalent to a

Answer	Marks	Guidance
constant.	3.1a	M1

Simplifies B s i n c o s e c  to B

Or

Answer	Marks	Guidance
Dcos2sec to Dcos	1.1a	M1

Completes reasoned argument

Answer	Marks	Guidance
to obtain c o s s e c   4 −	2.1	R1
Subtotal	4
Q	Marking instructions	AO

Answer	Marks
15(b)(i)	Explains that cosec is

undefined when c o s  = 1

Or

explains c o s  = 1 would mean

that s i n  = 0

or uses s i n   0 to show that

Answer	Marks	Guidance
cos=1should be rejected	2.4	E1
Subtotal	1
Q	Marking instructions	AO

Answer	Marks	Guidance
15(b)(ii)	Obtains 104.5, 255.5
CAO	2.2a	B1
Subtotal	1
Question 15 Total	6
Q	Marking instructions	AO

Question 15:
--- 15(a) ---
15(a) | Recalls the identity for
sin2=2sincosor an
identity for c o s  2 eg
cos2=2cos2−1
cos2=1−2sin2
c o s c o s s i n    2 = 2 − 2
This mark could be scored later
if compound angle formula is
used with a completely correct
argument. | 1.2 | B1 | sin2cosec+cos2sec
=2sincoscosec+ ( 2cos2−1 ) sec
=2sincoscosec+2cos2sec−sec
=2cos+2cos−sec
=4cos−sec
Substitutes A s i n c o s   and a
correct identity for cos2
OR
Substitutes s i n c o s   2 and an
identity for cos2with sign
errors condoned provided
cos2is not replaced with an
expression equivalent to a
constant. | 3.1a | M1
Simplifies B s i n c o s e c  to B
Or
Dcos2sec to Dcos | 1.1a | M1
Completes reasoned argument
to obtain c o s s e c   4 − | 2.1 | R1
Subtotal | 4
Q | Marking instructions | AO | Marks | Typical solution
--- 15(b)(i) ---
15(b)(i) | Explains that cosec is
undefined when c o s  = 1
Or
explains c o s  = 1 would mean
that s i n  = 0
or uses s i n   0 to show that
cos=1should be rejected | 2.4 | E1 | cos1 as cosec is undefined
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 15(b)(ii) ---
15(b)(ii) | Obtains 104.5, 255.5
CAO | 2.2a | B1 | =104.5 ,255.5
Subtotal | 1
Question 15 Total | 6
Q | Marking instructions | AO | Marks | Typical solution

Show LaTeX source

\begin{enumerate}[label=(\alph*)]
\item Show that the expression
$$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta$$
can be written as
$$4 \cos \theta - \sec \theta$$
where $\sin \theta \neq 0$ and $\cos \theta \neq 0$
[4 marks]

\item A student is attempting to solve the equation
$$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$

They use the result from part (a), and write the following incorrect solution:

$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3$

Step 1: $4 \cos \theta - \sec \theta = 3$

Step 2: $4 \cos \theta - \frac{1}{\cos \theta} - 3 = 0$

Step 3: $4 \cos^2 \theta - 3 \cos \theta - 1 = 0$

Step 4: $\cos \theta = 1$ or $\cos \theta = -0.25$

Step 5: $\theta = 0°, 104.5°, 255.5°, 360°$

\begin{enumerate}[label=(\roman*)]
\item Explain why the student should reject one of their values for $\cos \theta$ in Step 4.
[1 mark]

\item State the correct solutions to the equation
$$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$
[1 mark]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 1 2024 Q15 [6]}}

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