Edexcel PURE 2024 October — Question 9

Exam BoardEdexcel
ModulePURE
Year2024
SessionOctober
PaperDownload PDF ↗
TopicStandard trigonometric equations
TypeTransformed argument solving
DifficultyStandard +0.3 This is a standard A-level trigonometric equation requiring routine algebraic manipulation (converting tan to sin/cos, using Pythagorean identity) and solving a quadratic in sin θ, followed by a straightforward substitution and range adjustment. While it requires multiple steps and careful attention to the range, it involves only well-practiced techniques with no novel insight needed.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals
  1. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that the equation $$2 \tan \theta = 3 \cos \theta$$ can be written as $$3 \sin ^ { 2 } \theta + 2 \sin \theta - 3 = 0$$
  2. Hence solve, for \(- \pi < x < \pi\), the equation $$2 \tan \left( 2 x + \frac { \pi } { 3 } \right) = 3 \cos \left( 2 x + \frac { \pi } { 3 } \right)$$ giving your answers to 3 significant figures.
Question 9:
AnswerMarks
9(a)2sin
( 2tan=3cos) =3cos
AnswerMarks
cosM1
2 s i n ( ) 
3 c o s 2 s i n 3 c o s 2 3 1 s i n 2     =  = = −
AnswerMarks
c o s M1
( )
AnswerMarks
2sin=3 1−sin2 3sin2+2sin−3=0*A1*
(3)
AnswerMarks
(b)   −1 10
sin2x+  =  (May only see positive root)
  3  3
AnswerMarks
NB decimal roots are: −1.387…, 0.7207…M1
2x+ =sin−1(0.7207...) x=...
AnswerMarks
3M1
−0.121, −2.50, 0.645, 3.02A1A1
(4)

Total 7

M1: Attempts to solve the quadratic 3 s i n 2 x + 2 s i n x − 3 = 0 to obtain a value for sin x
where x is any variable. Usual rules apply for solving a quadratic (via a calculator is
also acceptable and may imply this mark). If no working is shown then the root(s)
must be correct but condone premature rounding e.g. 0.72, −1.3
M1: Attempts to find one angle within the range by finding the inverse sine of one of their
roots, subtracting and dividing by 2.
3
For this mark allow to work in degrees if done correctly. E.g. they would need to
change to 60° and then find the inverse sine of one of their roots in degrees,
3
subtract 60° and divide by 2.
May be implied by a correct value of x in degrees or radians.
NB the answers in degrees are: − 1 4 3 , − 6 . 9 4 , 3 6 . 9 , 1 7 3 (3sf)
46.1...−
Do not allow the mixing of degrees and radians for this mark e.g. x= 3
2
A1: Any two of awrt −0.12, −2.5, 0.64 or 0.65, 3.0 (Must be in radians)
A1: All four of awrt − 0 . 1 2 1 , − 2 . 5 0 , 0 . 6 4 5 , 3 . 0 2
PMT
and no others in the range.
(Must be in radians)
Condone −2.5 for −2.50 but the others must be awrt as shown for this final mark.
Question 9:
--- 9(a) ---
9(a) | 2sin
( 2tan=3cos) =3cos
cos | M1
2 s i n ( ) 
3 c o s 2 s i n 3 c o s 2 3 1 s i n 2     =  = = −
c o s  | M1
( )
2sin=3 1−sin2 3sin2+2sin−3=0* | A1*
(3)
(b) |    −1 10
sin2x+  =  (May only see positive root)
  3  3
NB decimal roots are: −1.387…, 0.7207… | M1

2x+ =sin−1(0.7207...) x=...
3 | M1
−0.121, −2.50, 0.645, 3.02 | A1A1
(4)
Total 7
M1: Attempts to solve the quadratic 3 s i n 2 x + 2 s i n x − 3 = 0 to obtain a value for sin x
where x is any variable. Usual rules apply for solving a quadratic (via a calculator is
also acceptable and may imply this mark). If no working is shown then the root(s)
must be correct but condone premature rounding e.g. 0.72, −1.3
M1: Attempts to find one angle within the range by finding the inverse sine of one of their

roots, subtracting and dividing by 2.
3
For this mark allow to work in degrees if done correctly. E.g. they would need to

change to 60° and then find the inverse sine of one of their roots in degrees,
3
subtract 60° and divide by 2.
May be implied by a correct value of x in degrees or radians.
NB the answers in degrees are: − 1 4 3 , − 6 . 9 4 , 3 6 . 9 , 1 7 3 (3sf)
46.1...−
Do not allow the mixing of degrees and radians for this mark e.g. x= 3
2
A1: Any two of awrt −0.12, −2.5, 0.64 or 0.65, 3.0 (Must be in radians)
A1: All four of awrt − 0 . 1 2 1 , − 2 . 5 0 , 0 . 6 4 5 , 3 . 0 2
PMT
and no others in the range.
(Must be in radians)
Condone −2.5 for −2.50 but the others must be awrt as shown for this final mark.
\begin{enumerate}
  \item In this question you must show detailed reasoning.
\end{enumerate}

Solutions relying entirely on calculator technology are not acceptable.\\
(a) Show that the equation

$$2 \tan \theta = 3 \cos \theta$$

can be written as

$$3 \sin ^ { 2 } \theta + 2 \sin \theta - 3 = 0$$

(b) Hence solve, for $- \pi < x < \pi$, the equation

$$2 \tan \left( 2 x + \frac { \pi } { 3 } \right) = 3 \cos \left( 2 x + \frac { \pi } { 3 } \right)$$

giving your answers to 3 significant figures.

\hfill \mbox{\textit{Edexcel PURE 2024 Q9}}
This paper (11 questions)
View full paper
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11