AQA AS Paper 2 Specimen — Question 7 5 marks

Exam BoardAQA
ModuleAS Paper 2 (AS Paper 2)
SessionSpecimen
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeProduct of trig functions
DifficultyStandard +0.3 This is a trigonometric equation requiring factorization and substitution techniques that are standard for AS-level. Students must factor out sin θ, use tan θ = sin θ/cos θ, then solve a quadratic in sin θ. While it involves multiple steps and careful algebraic manipulation, the techniques are all routine for this level with no novel insight required. The 5-mark allocation and restricted domain make it slightly above trivial but still below average difficulty.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals

Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]

Question 7:
AnswerMarks Guidance
7Divides or multiplies by cosθ AO3.1a
+2 =3
cosθ cosθ
tan2 θ + 2tan θ – 3 = 0
(tan θ + 3)(tan θ – 1) = 0
tan θ = 1 or –3
θ=45º or 108º
ALT
sinθtanθcosθ+2sinθcosθ=3cos2θ
sin2θ+2sinθcosθ−3cos2θ=0
(sinθ+3cosθ)(sinθ−cosθ)=0
tan θ = 1 or –3
θ=45º or 108º
AnswerMarks Guidance
Obtains correct quadraticAO1.1b A1
Applies a correct method to solve
AnswerMarks Guidance
‘their’ quadratic PIAO1.1a M1
Finds two correct values of tan θ
AnswerMarks Guidance
from ‘their’ quadraticAO1.1b A1F
Obtains two correct answers CAOAO1.1b A1
Total5
QMarking Instructions AO
Question 7:
7 | Divides or multiplies by cosθ | AO3.1a | M1 | sinθtanθ sinθ
+2 =3
cosθ cosθ
tan2 θ + 2tan θ – 3 = 0
(tan θ + 3)(tan θ – 1) = 0
tan θ = 1 or –3
θ=45º or 108º
ALT
sinθtanθcosθ+2sinθcosθ=3cos2θ
sin2θ+2sinθcosθ−3cos2θ=0
(sinθ+3cosθ)(sinθ−cosθ)=0
tan θ = 1 or –3
θ=45º or 108º
Obtains correct quadratic | AO1.1b | A1
Applies a correct method to solve
‘their’ quadratic PI | AO1.1a | M1
Finds two correct values of tan θ
from ‘their’ quadratic | AO1.1b | A1F
Obtains two correct answers CAO | AO1.1b | A1
Total | 5
Q | Marking Instructions | AO | Marks | Typical Solution
Solve the equation
$$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$

Give all values of $\theta$ to the nearest degree in the interval $0° < \theta < 180°$

Fully justify your answer.
[5 marks]

\hfill \mbox{\textit{AQA AS Paper 2  Q7 [5]}}